jcalm 2017 introduction to designs and steiner triple systems · 2/124 outline introduction to...
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
JCALM 2017Introduction to Designs and Steiner Triple Systems
Fionn Mc Inerney
Universite Cote d’Azur, Inria, CNRS, I3S, France
May 3, 2017
Fionn Mc Inerney JCALM 2017 Introduction to Designs and Steiner Triple Systems
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
JCALM 2017Decomposition of Complete Graphs
Fionn Mc Inerney
Universite Cote d’Azur, Inria, CNRS, I3S, France
May 3, 2017
Fionn Mc Inerney JCALM 2017 Decomposition of Complete Graphs
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
1 Introduction to Designs: Definitions and Examples
2 Steiner Triple Systems: Definition and Existence
3 Other t-designs, Affine and Projective planes, and Conclusion
Fionn Mc Inerney JCALM 2017 Decomposition of Complete Graphs
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
(7,3,1) design: Decompose K7 into K3
12
3
4
56
7
12
3
4
56
7
Decomposition: Partition of edges.
Fionn Mc Inerney JCALM 2017 Decomposition of Complete Graphs
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
(7,3,1) design: Decompose K7 into K3
12
3
4
56
7
12
3
4
56
7
Decomposition: Partition of edges.
Fionn Mc Inerney JCALM 2017 Decomposition of Complete Graphs
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
(7,3,1) design: Decompose K7 into K3
12
3
4
56
7
12
3
4
56
7
Decomposition: Partition of edges.
Fionn Mc Inerney JCALM 2017 Decomposition of Complete Graphs
7/124
OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
(7,3,1) design: Decompose K7 into K3
12
3
4
56
7
12
3
4
56
7
Decomposition: Partition of edges.
Fionn Mc Inerney JCALM 2017 Decomposition of Complete Graphs
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
(7,3,1) design: Decompose K7 into K3
12
3
4
56
7
12
3
4
56
7
Decomposition: Partition of edges.
Fionn Mc Inerney JCALM 2017 Decomposition of Complete Graphs
9/124
OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
(7,3,1) design: Decompose K7 into K3
12
3
4
56
7
12
3
4
56
7
Decomposition: Partition of edges.
Fionn Mc Inerney JCALM 2017 Decomposition of Complete Graphs
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
(7,3,1) design: Decompose K7 into K3
12
3
4
56
7
12
3
4
56
7
Decomposition: Partition of edges.
Fionn Mc Inerney JCALM 2017 Decomposition of Complete Graphs
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
(7,3,1) design: Decompose K7 into K3
12
3
4
56
7
12
3
4
56
7
Decomposition: Partition of edges.
Fionn Mc Inerney JCALM 2017 Decomposition of Complete Graphs
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
K4 and K5 don’t decompose into K3
K4 doesn’t decompose into K3:for all v ∈ V (K4) and u ∈ V (K3), deg(v) = 3 anddeg(u) = 2 but 2 - 3.
K5 doesn’t decompose into K3:|E (K5)| = 10 and |E (K3)| = 3 but 3 - 10.
Main question: when do designs exist?
Fionn Mc Inerney JCALM 2017 Decomposition of Complete Graphs
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
K4 and K5 don’t decompose into K3
K4 doesn’t decompose into K3:for all v ∈ V (K4) and u ∈ V (K3), deg(v) = 3 anddeg(u) = 2 but 2 - 3.
K5 doesn’t decompose into K3:|E (K5)| = 10 and |E (K3)| = 3 but 3 - 10.
Main question: when do designs exist?
Fionn Mc Inerney JCALM 2017 Decomposition of Complete Graphs
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
K4 and K5 don’t decompose into K3
K4 doesn’t decompose into K3:for all v ∈ V (K4) and u ∈ V (K3), deg(v) = 3 anddeg(u) = 2 but 2 - 3.
K5 doesn’t decompose into K3:|E (K5)| = 10 and |E (K3)| = 3 but 3 - 10.
Main question: when do designs exist?
Fionn Mc Inerney JCALM 2017 Decomposition of Complete Graphs
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
t-designs
Definition
A t − (n, k, λ) design is a set of subsets of size k (blocks) of a setV of n vertices such that all subsets of size t belong to exactly λblocks.
Abbreviated to (n, k , λ) design when t = 2.
Fionn Mc Inerney JCALM 2017 Decomposition of Complete Graphs
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
t-designs
Definition
A t − (n, k, λ) design is a set of subsets of size k (blocks) of a setV of n vertices such that all subsets of size t belong to exactly λblocks.
Abbreviated to (n, k , λ) design when t = 2.
Fionn Mc Inerney JCALM 2017 Decomposition of Complete Graphs
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
2-designs (t = 2) with λ = 1: Decompose Kn into Kk
Definition
An (n, k, 1) design is a set of subsets of size k (blocks) of a set Vof n vertices such that any pair of elements belongs to exactly 1block.
Definition
An (n, k , 1) design is a decomposition of Kn (edge partition) intoKk .
Each block of size k : Kk .
Any pair (t = 2) of elements in exactly one (λ = 1) block:each edge of Kn in exactly one Kk .
Fionn Mc Inerney JCALM 2017 Decomposition of Complete Graphs
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
2-designs (t = 2) with λ = 1: Decompose Kn into Kk
Definition
An (n, k, 1) design is a set of subsets of size k (blocks) of a set Vof n vertices such that any pair of elements belongs to exactly 1block.
Definition
An (n, k , 1) design is a decomposition of Kn (edge partition) intoKk .
Each block of size k : Kk .
Any pair (t = 2) of elements in exactly one (λ = 1) block:each edge of Kn in exactly one Kk .
Fionn Mc Inerney JCALM 2017 Decomposition of Complete Graphs
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
2-designs (t = 2) with λ = 1: Decompose Kn into Kk
Definition
An (n, k, 1) design is a set of subsets of size k (blocks) of a set Vof n vertices such that any pair of elements belongs to exactly 1block.
Definition
An (n, k , 1) design is a decomposition of Kn (edge partition) intoKk .
Each block of size k : Kk .
Any pair (t = 2) of elements in exactly one (λ = 1) block:each edge of Kn in exactly one Kk .
Fionn Mc Inerney JCALM 2017 Decomposition of Complete Graphs
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
2-designs (t = 2) with λ = 1: Decompose Kn into Kk
Definition
An (n, k, 1) design is a set of subsets of size k (blocks) of a set Vof n vertices such that any pair of elements belongs to exactly 1block.
Definition
An (n, k , 1) design is a decomposition of Kn (edge partition) intoKk .
Each block of size k : Kk .
Any pair (t = 2) of elements in exactly one (λ = 1) block:each edge of Kn in exactly one Kk .
Fionn Mc Inerney JCALM 2017 Decomposition of Complete Graphs
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
(4,2,1) design: Decompose K4 into K2
1
2
3
4
(n, 2, 1) designs (k = 2) trivially exist (decomposing Kn into K2).
Fionn Mc Inerney JCALM 2017 Decomposition of Complete Graphs
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
(7,3,1) design: Decompose K7 into K3
12
3
4
56
7
01
2
34
5
61
2
3
4
56
7
01
2
34
5
6
Decomposition obtained by rotation of K3.Bi = {{i , i + 2, i + 3} | i ∈ [7]} (indices (mod 7)).
Fionn Mc Inerney JCALM 2017 Decomposition of Complete Graphs
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
(7,3,1) design: Decompose K7 into K3
12
3
4
56
7
01
2
34
5
61
2
3
4
56
7
01
2
34
5
6
Decomposition obtained by rotation of K3.Bi = {{i , i + 2, i + 3} | i ∈ [7]} (indices (mod 7)).
Fionn Mc Inerney JCALM 2017 Decomposition of Complete Graphs
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
(7,3,1) design: Decompose K7 into K3
12
3
4
56
7
01
2
34
5
61
2
3
4
56
7
01
2
34
5
6
Decomposition obtained by rotation of K3.Bi = {{i , i + 2, i + 3} | i ∈ [7]} (indices (mod 7)).
Fionn Mc Inerney JCALM 2017 Decomposition of Complete Graphs
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
(7,3,1) design: Decompose K7 into K3
12
3
4
56
7
01
2
34
5
61
2
3
4
56
7
01
2
34
5
6
Decomposition obtained by rotation of K3.Bi = {{i , i + 2, i + 3} | i ∈ [7]} (indices (mod 7)).
Fionn Mc Inerney JCALM 2017 Decomposition of Complete Graphs
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
(7,3,1) design: Decompose K7 into K3
12
3
4
56
7
01
2
34
5
61
2
3
4
56
7
01
2
34
5
6
Decomposition obtained by rotation of K3.Bi = {{i , i + 2, i + 3} | i ∈ [7]} (indices (mod 7)).
Fionn Mc Inerney JCALM 2017 Decomposition of Complete Graphs
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
(7,3,1) design: Decompose K7 into K3
12
3
4
56
7
01
2
34
5
61
2
3
4
56
7
01
2
34
5
6
Decomposition obtained by rotation of K3.Bi = {{i , i + 2, i + 3} | i ∈ [7]} (indices (mod 7)).
Fionn Mc Inerney JCALM 2017 Decomposition of Complete Graphs
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
(7,3,1) design: Decompose K7 into K3
12
3
4
56
7
01
2
34
5
61
2
3
4
56
7
01
2
34
5
6
Decomposition obtained by rotation of K3.Bi = {{i , i + 2, i + 3} | i ∈ [7]} (indices (mod 7)).
Fionn Mc Inerney JCALM 2017 Decomposition of Complete Graphs
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
(7,3,1) design: Decompose K7 into K3
12
3
4
56
7
01
2
34
5
61
2
3
4
56
7
01
2
34
5
6
Decomposition obtained by rotation of K3.Bi = {{i , i + 2, i + 3} | i ∈ [7]} (indices (mod 7)).
Fionn Mc Inerney JCALM 2017 Decomposition of Complete Graphs
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
(13,3,1) design: Decompose K13 into K3
Decomposition obtained by rotation of two K3.Bi = {{i , i + 2, i + 7} | i ∈ [13]} (indices (mod 13)).Bj = {{j , j + 1, j + 4} | j ∈ [13]} (indices (mod 13)).
Fionn Mc Inerney JCALM 2017 Decomposition of Complete Graphs
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
(13,4,1) design: Decompose K13 into K4
Decomposition obtained by rotation of K4.Bi = {{i , i + 4, i + 5, i + 7} | i ∈ [13]} (indices (mod 13)).
Fionn Mc Inerney JCALM 2017 Decomposition of Complete Graphs
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
(9,3,1) design: Decompose K9 into K3
1 2 3
4 5 6
7 8 9
Fionn Mc Inerney JCALM 2017 Decomposition of Complete Graphs
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
(9,3,1) design: Decompose K9 into K3
1 2 3
4 5 6
7 8 9
Blue blocks partition vertices.
Fionn Mc Inerney JCALM 2017 Decomposition of Complete Graphs
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
(9,3,1) design: Decompose K9 into K3
1 2 3
4 5 6
7 8 9
Red blocks partition vertices.
Fionn Mc Inerney JCALM 2017 Decomposition of Complete Graphs
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
(9,3,1) design: Decompose K9 into K3
1 2 3
4 5 6
7 8 9
Green blocks partition vertices.
Fionn Mc Inerney JCALM 2017 Decomposition of Complete Graphs
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
(9,3,1) design: Decompose K9 into K3
1 2 3
4 5 6
7 8 9
Gray blocks partition vertices.
Fionn Mc Inerney JCALM 2017 Decomposition of Complete Graphs
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
(9,3,1) design: Decompose K9 into K3
1 2 3
4 5 6
7 8 9
Fionn Mc Inerney JCALM 2017 Decomposition of Complete Graphs
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
(9,3,1) design: Decompose K9 into K3
Fionn Mc Inerney JCALM 2017 Decomposition of Complete Graphs
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Resolvable designs
Definition
A design is resolvable if the blocks (Kk) can be grouped into setsthat partition the vertices.
Fionn Mc Inerney JCALM 2017 Decomposition of Complete Graphs
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Resolvable (4,2,1) design:Resolvable decomposition of K4 into K2
1
2
3
4
Definition
A design is resolvable if theblocks (Kk) can be grouped intosets that partition the vertices.
For n even, Kn decomposes into n − 1 perfect matchings.
Fionn Mc Inerney JCALM 2017 Decomposition of Complete Graphs
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Resolvable (4,2,1) design:Resolvable decomposition of K4 into K2
1
2
3
4
Definition
A design is resolvable if theblocks (Kk) can be grouped intosets that partition the vertices.
For n even, Kn decomposes into n − 1 perfect matchings.
Fionn Mc Inerney JCALM 2017 Decomposition of Complete Graphs
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Resolvable (4,2,1) design:Resolvable decomposition of K4 into K2
1
2
3
4
Definition
A design is resolvable if theblocks (Kk) can be grouped intosets that partition the vertices.
For n even, Kn decomposes into n − 1 perfect matchings.
Fionn Mc Inerney JCALM 2017 Decomposition of Complete Graphs
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Resolvable (4,2,1) design:Resolvable decomposition of K4 into K2
1
2
3
4
Definition
A design is resolvable if theblocks (Kk) can be grouped intosets that partition the vertices.
For n even, Kn decomposes into n − 1 perfect matchings.
Fionn Mc Inerney JCALM 2017 Decomposition of Complete Graphs
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Kirkman’s Schoolgirl Problem:Resolvable decomposition of K15 into K3
Kirkman’s Schoolgirl Problem(Reverend Kirkman, 1850)
15 young ladies in a school walk out 3abreast for 7 days in succession. Is itpossible to arrange them daily, so that no2 walk twice abreast?
Seeked actual solution for resolvable(15,3,1) design.
Same can be applied to Coati team.
21 3
54 6
87 9
1110 12
1413 15
Fionn Mc Inerney JCALM 2017 Decomposition of Complete Graphs
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Kirkman’s Schoolgirl Problem:Resolvable decomposition of K15 into K3
Kirkman’s Schoolgirl Problem(Reverend Kirkman, 1850)
15 young ladies in a school walk out 3abreast for 7 days in succession. Is itpossible to arrange them daily, so that no2 walk twice abreast?
Seeked actual solution for resolvable(15,3,1) design.
Same can be applied to Coati team.
21 3
54 6
87 9
1110 12
1413 15
Fionn Mc Inerney JCALM 2017 Decomposition of Complete Graphs
38/124
OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Kirkman’s Schoolgirl Problem:Resolvable decomposition of K15 into K3
Kirkman’s Schoolgirl Problem(Reverend Kirkman, 1850)
15 young ladies in a school walk out 3abreast for 7 days in succession. Is itpossible to arrange them daily, so that no2 walk twice abreast?
Seeked actual solution for resolvable(15,3,1) design.
Same can be applied to Coati team.
21 3
54 6
87 9
1110 12
1413 15
Fionn Mc Inerney JCALM 2017 Decomposition of Complete Graphs
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Coati Schoolgirl Problem - Day 1
Jeanne-ClaudiaMichelle Christine
Frederica V.Nicole N. Patricia
AndreaFiona Stephanie
Nicole H.Juliana Joanna
Frederica G.Christelle Valentina
Fionn Mc Inerney JCALM 2017 Decomposition of Complete Graphs
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Coati Schoolgirl Problem - Day 1
Jeanne-ClaudiaMichelle Christine
Frederica V.Nicole N. Patricia
AndreaFiona Stephanie
Nicole H.Juliana Joanna
Frederica G.Christelle Valentina
Fionn Mc Inerney JCALM 2017 Decomposition of Complete Graphs
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Coati Schoolgirl Problem - Day 1
Jeanne-ClaudiaMichelle Christine
Frederica V.Nicole N. Patricia
AndreaFiona Stephanie
Nicole H.Juliana Joanna
Frederica G.Christelle Valentina
Fionn Mc Inerney JCALM 2017 Decomposition of Complete Graphs
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Coati Schoolgirl Problem - Day 2
Nicole N.Michelle Christelle
JulianaFiona Frederica V.
Nicole H.Andrea Patricia
ChristineFrederica G. Joanna
ValentinaStephanie Jeanne-Claudia
Fionn Mc Inerney JCALM 2017 Decomposition of Complete Graphs
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Coati Schoolgirl Problem - Day 2
Nicole N.Michelle Christelle
JulianaFiona Frederica V.
Nicole H.Andrea Patricia
ChristineFrederica G. Joanna
ValentinaStephanie Jeanne-Claudia
Fionn Mc Inerney JCALM 2017 Decomposition of Complete Graphs
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Coati Schoolgirl Problem - Day 3
FionaNicole N. Jeanne-Claudia
ChristelleJuliana Andrea
Frederica G.Nicole H. Stephanie
PatriciaChristine Valentina
MichelleJoanna Frederica V.
Fionn Mc Inerney JCALM 2017 Decomposition of Complete Graphs
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Coati Schoolgirl Problem - Day 3
FionaNicole N. Jeanne-Claudia
ChristelleJuliana Andrea
Frederica G.Nicole H. Stephanie
PatriciaChristine Valentina
MichelleJoanna Frederica V.
Fionn Mc Inerney JCALM 2017 Decomposition of Complete Graphs
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Coati Schoolgirl Problem - Day 4
Jeanne-ClaudiaChristelle Nicole H.
AndreaFrederica V. Christine
StephaniePatricia Michelle
ValentinaJoanna Fiona
JulianaNicole N. Frederica G.
Fionn Mc Inerney JCALM 2017 Decomposition of Complete Graphs
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Coati Schoolgirl Problem - Day 4
Jeanne-ClaudiaChristelle Nicole H.
AndreaFrederica V. Christine
StephaniePatricia Michelle
ValentinaJoanna Fiona
JulianaNicole N. Frederica G.
Fionn Mc Inerney JCALM 2017 Decomposition of Complete Graphs
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Coati Schoolgirl Problem - Day 5
ChristelleFiona Christine
Jeanne-ClaudiaJuliana Patricia
Nicole H.Frederica V. Valentina
Frederica G.Andrea Michelle
JoannaStephanie Nicole N.
Fionn Mc Inerney JCALM 2017 Decomposition of Complete Graphs
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Coati Schoolgirl Problem - Day 5
ChristelleFiona Christine
Jeanne-ClaudiaJuliana Patricia
Nicole H.Frederica V. Valentina
Frederica G.Andrea Michelle
JoannaStephanie Nicole N.
Fionn Mc Inerney JCALM 2017 Decomposition of Complete Graphs
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Coati Schoolgirl Problem - Day 6
Frederica V.Christelle Stephanie
AndreaJeanne-Claudia Joanna
ChristineNicole H. Nicole N.
PatriciaFrederica G. Fiona
MichelleValentina Juliana
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Coati Schoolgirl Problem - Day 6
Frederica V.Christelle Stephanie
AndreaJeanne-Claudia Joanna
ChristineNicole H. Nicole N.
PatriciaFrederica G. Fiona
MichelleValentina Juliana
Fionn Mc Inerney JCALM 2017 Decomposition of Complete Graphs
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Coati Schoolgirl Problem - Day 7
StephanieChristine Juliana
JoannaPatricia Christelle
Nicole N.Valentina Andrea
FionaMichelle Nicole H.
Frederica V.Jeanne-Claudia Frederica G.
Fionn Mc Inerney JCALM 2017 Decomposition of Complete Graphs
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Coati Schoolgirl Problem - Day 7
StephanieChristine Juliana
JoannaPatricia Christelle
Nicole N.Valentina Andrea
FionaMichelle Nicole H.
Frederica V.Jeanne-Claudia Frederica G.
Fionn Mc Inerney JCALM 2017 Decomposition of Complete Graphs
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Necessary condition (1) for an (n, k , 1) design:Decomposition of Kn into Kk
Lemma
If an (n, k , 1) design exists, then n(n − 1) ≡ 0 (mod k(k − 1)).
Proof.
|E (Kn)| = n(n−1)2 must be a multiple of |E (Kk)| = k(k−1)
2 .
Necessary Condition (1)
n(n − 1) ≡ 0 (mod k(k − 1)).
K5 doesn’t decompose into K3.
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Necessary condition (1) for an (n, k , 1) design:Decomposition of Kn into Kk
Lemma
If an (n, k , 1) design exists, then n(n − 1) ≡ 0 (mod k(k − 1)).
Proof.
|E (Kn)| = n(n−1)2 must be a multiple of |E (Kk)| = k(k−1)
2 .
Necessary Condition (1)
n(n − 1) ≡ 0 (mod k(k − 1)).
K5 doesn’t decompose into K3.
Fionn Mc Inerney JCALM 2017 Decomposition of Complete Graphs
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Necessary condition (1) for an (n, k , 1) design:Decomposition of Kn into Kk
Lemma
If an (n, k , 1) design exists, then n(n − 1) ≡ 0 (mod k(k − 1)).
Proof.
|E (Kn)| = n(n−1)2 must be a multiple of |E (Kk)| = k(k−1)
2 .
Necessary Condition (1)
n(n − 1) ≡ 0 (mod k(k − 1)).
K5 doesn’t decompose into K3.
Fionn Mc Inerney JCALM 2017 Decomposition of Complete Graphs
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Necessary condition (1) for an (n, k , 1) design:Decomposition of Kn into Kk
Lemma
If an (n, k , 1) design exists, then n(n − 1) ≡ 0 (mod k(k − 1)).
Proof.
|E (Kn)| = n(n−1)2 must be a multiple of |E (Kk)| = k(k−1)
2 .
Necessary Condition (1)
n(n − 1) ≡ 0 (mod k(k − 1)).
K5 doesn’t decompose into K3.
Fionn Mc Inerney JCALM 2017 Decomposition of Complete Graphs
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Necessary condition (2) for an (n, k , 1) design:Decomposition of Kn into Kk
Lemma
If an (n, k , 1) design exists, then n − 1 ≡ 0 (mod k − 1).
Proof.
For all v ∈ V (Kn) and u ∈ V (Kk), deg(v) = n − 1 must be amultiple of deg(u) = k − 1.
Necessary condition (2)
n − 1 ≡ 0 (mod k − 1).
K4 doesn’t decompose into K3.
Fionn Mc Inerney JCALM 2017 Decomposition of Complete Graphs
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Necessary condition (2) for an (n, k , 1) design:Decomposition of Kn into Kk
Lemma
If an (n, k , 1) design exists, then n − 1 ≡ 0 (mod k − 1).
Proof.
For all v ∈ V (Kn) and u ∈ V (Kk), deg(v) = n − 1 must be amultiple of deg(u) = k − 1.
Necessary condition (2)
n − 1 ≡ 0 (mod k − 1).
K4 doesn’t decompose into K3.
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Necessary condition (2) for an (n, k , 1) design:Decomposition of Kn into Kk
Lemma
If an (n, k , 1) design exists, then n − 1 ≡ 0 (mod k − 1).
Proof.
For all v ∈ V (Kn) and u ∈ V (Kk), deg(v) = n − 1 must be amultiple of deg(u) = k − 1.
Necessary condition (2)
n − 1 ≡ 0 (mod k − 1).
K4 doesn’t decompose into K3.
Fionn Mc Inerney JCALM 2017 Decomposition of Complete Graphs
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Necessary condition (2) for an (n, k , 1) design:Decomposition of Kn into Kk
Lemma
If an (n, k , 1) design exists, then n − 1 ≡ 0 (mod k − 1).
Proof.
For all v ∈ V (Kn) and u ∈ V (Kk), deg(v) = n − 1 must be amultiple of deg(u) = k − 1.
Necessary condition (2)
n − 1 ≡ 0 (mod k − 1).
K4 doesn’t decompose into K3.
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Necessary Conditions are Sufficient in General
Necessary Condition (1)
n(n − 1) ≡ 0 (mod k(k − 1)).
Necessary condition (2)
n − 1 ≡ 0 (mod k − 1).
In general, for n large enough, these 2 conditions are sufficient.
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Necessary Conditions are Sufficient in General
Necessary Condition (1)
n(n − 1) ≡ 0 (mod k(k − 1)).
Necessary condition (2)
n − 1 ≡ 0 (mod k − 1).
In general, for n large enough, these 2 conditions are sufficient.
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Steiner Triple Systems: Decomposition of Kn into K3
Definition
A Steiner Triple System is an (n, 3, 1) design.
Steiner asked the question if they exist in 1853, withoutknowing Kirkman had already proved their existence in 1847.
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Steiner Triple Systems: Decomposition of Kn into K3
Definition
A Steiner Triple System is an (n, 3, 1) design.
Steiner asked the question if they exist in 1853, withoutknowing Kirkman had already proved their existence in 1847.
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Necessary conditions for Steiner Triple Systems:Decomposition of Kn into K3
Lemma
If an (n, 3, 1) design exists, then n ≡ 1 or 3 (mod 6).
Proof.
NC(2) ⇒ n − 1 ≡ 0 (mod 2) ⇒ n odd ⇒ n ≡ 1, 3 or 5(mod 6).
NC(1) ⇒ n(n− 1) ≡ 0 (mod 6) ⇒ n ≡ 1 or 3 (mod 6) since
if n = 6k + 5, then n(n−1)6 = 6k+5(6k+4)
6 = 18k2+27k+103 /∈ Z.
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Necessary conditions for Steiner Triple Systems:Decomposition of Kn into K3
Lemma
If an (n, 3, 1) design exists, then n ≡ 1 or 3 (mod 6).
Proof.
NC(2) ⇒ n − 1 ≡ 0 (mod 2) ⇒ n odd ⇒ n ≡ 1, 3 or 5(mod 6).
NC(1) ⇒ n(n− 1) ≡ 0 (mod 6) ⇒ n ≡ 1 or 3 (mod 6) since
if n = 6k + 5, then n(n−1)6 = 6k+5(6k+4)
6 = 18k2+27k+103 /∈ Z.
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Necessary conditions for Steiner Triple Systems:Decomposition of Kn into K3
Lemma
If an (n, 3, 1) design exists, then n ≡ 1 or 3 (mod 6).
Proof.
NC(2) ⇒ n − 1 ≡ 0 (mod 2) ⇒ n odd ⇒ n ≡ 1, 3 or 5(mod 6).
NC(1) ⇒ n(n− 1) ≡ 0 (mod 6) ⇒ n ≡ 1 or 3 (mod 6) since
if n = 6k + 5, then n(n−1)6 = 6k+5(6k+4)
6 = 18k2+27k+103 /∈ Z.
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Necessary Conditions are Sufficient for Steiner TripleSystems: Decomposition of Kn into K3
Theorem (Kirkman, 1847)
An (n, 3, 1) design exists ⇔ n ≡ 1 or 3 (mod 6).
i.e. Kn can be decomposed into K3 ⇔ n ≡ 1 or 3 (mod 6).
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Lemma 1: K2m+1 dec. K3 ⇔ Km∗2 dec. K3
Lemma 1
K2m+1 can be decomposed into K3 ⇔K2,··· ,2 = Km∗2 can be decomposed into K3.
1
1
1
1
1
1
1
1
1
1
1 z
. . .2m
Fionn Mc Inerney JCALM 2017 Decomposition of Complete Graphs
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Lemma 1: K2m+1 dec. K3 ⇔ Km∗2 dec. K3
Lemma 1
K2m+1 can be decomposed into K3 ⇔K2,··· ,2 = Km∗2 can be decomposed into K3.
1
1
1
1
1
1
1
1
1
1
1 z
. . .2m
Fionn Mc Inerney JCALM 2017 Decomposition of Complete Graphs
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Lemma 1: K2m+1 dec. K3 ⇔ Km∗2 dec. K3
Lemma 1
K2m+1 can be decomposed into K3 ⇔K2,··· ,2 = Km∗2 can be decomposed into K3.
1
1
1
1
1
1
1
1
1
1
1 z
. . .2m
Fionn Mc Inerney JCALM 2017 Decomposition of Complete Graphs
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Lemma 1: K2m+1 dec. K3 ⇔ Km∗2 dec. K3
Lemma 1
K2m+1 can be decomposed into K3 ⇔K2,··· ,2 = Km∗2 can be decomposed into K3.
1
1
1
1
1
1
1
1
1
1
. . .2m
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Lemma 1: K2m+1 dec. K3 ⇔ Km∗2 dec. K3
Lemma 1
K2m+1 can be decomposed into K3 ⇔K2,··· ,2 = Km∗2 can be decomposed into K3.
1
1
1
1
1
1
1
1
1
1
1 z
. . .2m
Fionn Mc Inerney JCALM 2017 Decomposition of Complete Graphs
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Lemma 1: K2m+1 dec. K3 ⇔ Km∗2 dec. K3
Lemma 1
K2m+1 can be decomposed into K3 ⇔K2,··· ,2 = Km∗2 can be decomposed into K3.
1
1
1
1
1
1
1
1
1
1
1 z
. . .2m
Fionn Mc Inerney JCALM 2017 Decomposition of Complete Graphs
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Proof of Lemma 1: K2m+1 dec. K3 ⇔ Km∗2 dec. K3
Proof.
Let V (K2m+1) = {v1, · · · , v2m+1}, D be a decomposition ofK2m+1 into K3, and A be the set of all blocks containingv2m+1.
|A| = m so w.l.o.g. suppose A = {vi , vi+m, v2m+1 | i ∈ [m]}.Then, D \ A is a decomposition of H into K3 where H is thegraph K2m with the edges vivi+m, i ∈ [m] removed. H isisomorphic to Km∗2.
In the other direction, if D ′ is a decomposition of H into K3,then D ′ ∪ A is a decomposition of K2m+1 into K3.
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Proof of Lemma 1: K2m+1 dec. K3 ⇔ Km∗2 dec. K3
Proof.
Let V (K2m+1) = {v1, · · · , v2m+1}, D be a decomposition ofK2m+1 into K3, and A be the set of all blocks containingv2m+1.
|A| = m so w.l.o.g. suppose A = {vi , vi+m, v2m+1 | i ∈ [m]}.Then, D \ A is a decomposition of H into K3 where H is thegraph K2m with the edges vivi+m, i ∈ [m] removed. H isisomorphic to Km∗2.
In the other direction, if D ′ is a decomposition of H into K3,then D ′ ∪ A is a decomposition of K2m+1 into K3.
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Proof of Lemma 1: K2m+1 dec. K3 ⇔ Km∗2 dec. K3
Proof.
Let V (K2m+1) = {v1, · · · , v2m+1}, D be a decomposition ofK2m+1 into K3, and A be the set of all blocks containingv2m+1.
|A| = m so w.l.o.g. suppose A = {vi , vi+m, v2m+1 | i ∈ [m]}.Then, D \ A is a decomposition of H into K3 where H is thegraph K2m with the edges vivi+m, i ∈ [m] removed. H isisomorphic to Km∗2.
In the other direction, if D ′ is a decomposition of H into K3,then D ′ ∪ A is a decomposition of K2m+1 into K3.
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Theorem 1: K3∗q dec. K3
Theorem 1
Kq,q,q = K3∗q can be decomposed into K3.
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
q q q
aibj
ci+j
Fionn Mc Inerney JCALM 2017 Decomposition of Complete Graphs
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Theorem 1: K3∗q dec. K3
Theorem 1
Kq,q,q = K3∗q can be decomposed into K3.
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
q q q
aibj
ci+j
Fionn Mc Inerney JCALM 2017 Decomposition of Complete Graphs
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Theorem 1: K3∗q dec. K3
Theorem 1
Kq,q,q = K3∗q can be decomposed into K3.
Example q = 2:
1
1
1
1
1
1
a0
a1
b0
b1
c0
c1
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Theorem 1: K3∗q dec. K3
Theorem 1
Kq,q,q = K3∗q can be decomposed into K3.
Example q = 2:
1
1
1
1
1
1
a0
a1
b0
b1
c0
c1
Fionn Mc Inerney JCALM 2017 Decomposition of Complete Graphs
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Theorem 1: K3∗q dec. K3
Theorem 1
Kq,q,q = K3∗q can be decomposed into K3.
Example q = 2:
1
1
1
1
1
1
a0
a1
b0
b1
c0
c1
Fionn Mc Inerney JCALM 2017 Decomposition of Complete Graphs
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Theorem 1: K3∗q dec. K3
Theorem 1
Kq,q,q = K3∗q can be decomposed into K3.
Example q = 2:
1
1
1
1
1
1
a0
a1
b0
b1
c0
c1
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Theorem 1: K3∗q dec. K3
Theorem 1
Kq,q,q = K3∗q can be decomposed into K3.
Example q = 2:
1
1
1
1
1
1
a0
a1
b0
b1
c0
c1
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Theorem 1: K3∗q dec. K3
Theorem 1
Kq,q,q = K3∗q can be decomposed into K3.
Example q = 2:
1
1
1
1
1
1
a0
a1
b0
b1
c0
c1
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Proof of Theorem 1: K3∗q dec. K3
Proof.
Let (A,B,C ) be the tripartition of K3∗q with A = {ai | i ∈ [q]}and B and C defined analogously. {{ai , bj , ci+j} | i ∈ [q], j ∈ [q]}(indices (mod q)) is a decomposition of K3∗q into K3.
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Theorem 2: ∀q ≥ 3, q 6= 6, K4∗q dec. K4
Theorem 2
For all q ≥ 3, q 6= 6, Kq,q,q,q = K4∗q can be decomposed into K4.
Proof for q odd.
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
q q q q
aibj
ci+j
ci−j
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Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Theorem 2: ∀q ≥ 3, q 6= 6, K4∗q dec. K4
Theorem 2
For all q ≥ 3, q 6= 6, Kq,q,q,q = K4∗q can be decomposed into K4.
Proof for q odd.
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
q q q q
aibj
ci+j
ci−j
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Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Theorem 2: ∀q ≥ 3, q 6= 6, K4∗q dec. K4
Theorem 2
For all q ≥ 3, q 6= 6, Kq,q,q,q = K4∗q can be decomposed into K4.
Example q = 3:
1
1
1
1
1
1
1
1
1
1
1
1a0
a1
a2
b0
b1
b2
c0
c1
c2
d0
d1
d2
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Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Theorem 2: ∀q ≥ 3, q 6= 6, K4∗q dec. K4
Theorem 2
For all q ≥ 3, q 6= 6, Kq,q,q,q = K4∗q can be decomposed into K4.
Example q = 3:
1
1
1
1
1
1
1
1
1
1
1
1a0
a1
a2
b0
b1
b2
c0
c1
c2
d0
d1
d2
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Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Theorem 2: ∀q ≥ 3, q 6= 6, K4∗q dec. K4
Theorem 2
For all q ≥ 3, q 6= 6, Kq,q,q,q = K4∗q can be decomposed into K4.
Example q = 3:
1
1
1
1
1
1
1
1
1
1
1
1a0
a1
a2
b0
b1
b2
c0
c1
c2
d0
d1
d2
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Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Theorem 2: ∀q ≥ 3, q 6= 6, K4∗q dec. K4
Theorem 2
For all q ≥ 3, q 6= 6, Kq,q,q,q = K4∗q can be decomposed into K4.
Example q = 3:
1
1
1
1
1
1
1
1
1
1
1
1a0
a1
a2
b0
b1
b2
c0
c1
c2
d0
d1
d2
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Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Theorem 2: ∀q ≥ 3, q 6= 6, K4∗q dec. K4
Theorem 2
For all q ≥ 3, q 6= 6, Kq,q,q,q = K4∗q can be decomposed into K4.
Example q = 3:
1
1
1
1
1
1
1
1
1
1
1
1a0
a1
a2
b0
b1
b2
c0
c1
c2
d0
d1
d2
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Theorem 2: ∀q ≥ 3, q 6= 6, K4∗q dec. K4
Theorem 2
For all q ≥ 3, q 6= 6, Kq,q,q,q = K4∗q can be decomposed into K4.
Example q = 3:
1
1
1
1
1
1
1
1
1
1
1
1a0
a1
a2
b0
b1
b2
c0
c1
c2
d0
d1
d2
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Theorem 2: ∀q ≥ 3, q 6= 6, K4∗q dec. K4
Theorem 2
For all q ≥ 3, q 6= 6, Kq,q,q,q = K4∗q can be decomposed into K4.
Example q = 3:
1
1
1
1
1
1
1
1
1
1
1
1a0
a1
a2
b0
b1
b2
c0
c1
c2
d0
d1
d2
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Theorem 2: ∀q ≥ 3, q 6= 6, K4∗q dec. K4
Theorem 2
For all q ≥ 3, q 6= 6, Kq,q,q,q = K4∗q can be decomposed into K4.
Example q = 3:
1
1
1
1
1
1
1
1
1
1
1
1a0
a1
a2
b0
b1
b2
c0
c1
c2
d0
d1
d2
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Theorem 2: ∀q ≥ 3, q 6= 6, K4∗q dec. K4
Theorem 2
For all q ≥ 3, q 6= 6, Kq,q,q,q = K4∗q can be decomposed into K4.
Example q = 3:
1
1
1
1
1
1
1
1
1
1
1
1a0
a1
a2
b0
b1
b2
c0
c1
c2
d0
d1
d2
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Theorem 2: ∀q ≥ 3, q 6= 6, K4∗q dec. K4
Theorem 2
For all q ≥ 3, q 6= 6, Kq,q,q,q = K4∗q can be decomposed into K4.
Example q = 3:
1
1
1
1
1
1
1
1
1
1
1
1a0
a1
a2
b0
b1
b2
c0
c1
c2
d0
d1
d2
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Theorem 2: ∀q ≥ 3, q 6= 6, K4∗q dec. K4
Theorem 2
For all q ≥ 3, q 6= 6, Kq,q,q,q = K4∗q can be decomposed into K4.
Example q = 3:
1
1
1
1
1
1
1
1
1
1
1
1a0
a1
a2
b0
b1
b2
c0
c1
c2
d0
d1
d2
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Proof of Theorem 2: ∀q ≥ 3, q 6= 6, K4∗q dec. K4
11111
11111
11111
11111
q q q q
aibj
ci+j
ci−j
Proof. (q odd)
(i , j) and (i ′, j ′) don’t exist such that i + j = i ′ + j ′ (mod q) andi − j = i ′ − j ′ (mod q). Such a pair exists ⇒ 2i = 2i ′ (mod q)and 2j = 2j ′ (mod q) ⇒ i = i ′ and j = j ′ since q is odd.
Doesn’t work when q is even: (i , j) and (i + q2 , j + q
2 ).(0, 0, 0, 0) and (q2 ,
q2 , 0, 0).
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Proof of Theorem 2: ∀q ≥ 3, q 6= 6, K4∗q dec. K4
11111
11111
11111
11111
q q q q
aibj
ci+j
ci−j
Proof. (q odd)
(i , j) and (i ′, j ′) don’t exist such that i + j = i ′ + j ′ (mod q) andi − j = i ′ − j ′ (mod q). Such a pair exists ⇒ 2i = 2i ′ (mod q)and 2j = 2j ′ (mod q) ⇒ i = i ′ and j = j ′ since q is odd.
Doesn’t work when q is even: (i , j) and (i + q2 , j + q
2 ).(0, 0, 0, 0) and (q2 ,
q2 , 0, 0).
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Theorem 3: For r ≡ 0 or 1 (mod 3), Kr dec. K3,K4,K6
Theorem 3
If r ≡ 0 or 1 (mod 3), then Kr can be decomposed into acombination of K3,K4, and K6.
Proof.
We prove by induction. The result is trivially true for r = 3, 4, 6and r = 7 we already saw. Suppose r = 9m + j with m ∈ Z+ andj ∈ {0, 1, 3, 4, 6, 7}.
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Theorem 3: For r ≡ 0 or 1 (mod 3), Kr dec. K3,K4,K6
Theorem 3
If r ≡ 0 or 1 (mod 3), then Kr can be decomposed into acombination of K3,K4, and K6.
Proof.
We prove by induction. The result is trivially true for r = 3, 4, 6and r = 7 we already saw. Suppose r = 9m + j with m ∈ Z+ andj ∈ {0, 1, 3, 4, 6, 7}.
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Theorem 3: For r ≡ 0 or 1 (mod 3), Kr dec. K3,K4,K6
Example: r = 9.
1
1
1
1
1
1
1
1
1
K9 = 3K3 ∪ K3,3,3
1
1
1
1
1
1
1
1
1
K3 K3 K3
Base Case: K3 dec. K3,K4,K6.
1
1
1
1
1
1
1
1
1
K3,3,3
Thm 1: K3∗q dec. K3.
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Theorem 3: For r ≡ 0 or 1 (mod 3), Kr dec. K3,K4,K6
Example: r = 9.
1
1
1
1
1
1
1
1
1
K9 = 3K3 ∪ K3,3,3
1
1
1
1
1
1
1
1
1
K3 K3 K3
Base Case: K3 dec. K3,K4,K6.
1
1
1
1
1
1
1
1
1
K3,3,3
Thm 1: K3∗q dec. K3.
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Theorem 3: For r ≡ 0 or 1 (mod 3), Kr dec. K3,K4,K6
r = 9m + j for m ∈ Z+ and j ∈ {0, 1, 3, 4, 6, 7}.
1
1
1
1
1
1
1
1
1
K9m = 3K3m ∪ K3m,3m,3m
111
111
111
K3m K3m K3m
I.H.: K3m dec. K3,K4,K6.
111
111
111
K3m,3m,3m
Thm 1: K3∗q dec. K3.
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Theorem 3: For r ≡ 0 or 1 (mod 3), Kr dec. K3,K4,K6
r = 9m + j for m ∈ Z+ and j ∈ {0, 1, 3, 4, 6, 7}.
1
1
1
1
1
1
1
1
1
K9m = 3K3m ∪ K3m,3m,3m
111
111
111
K3m K3m K3m
I.H.: K3m dec. K3,K4,K6.
111
111
111
K3m,3m,3m
Thm 1: K3∗q dec. K3.
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Theorem 3: For r ≡ 0 or 1 (mod 3), Kr dec. K3,K4,K6
r = 9m + j for m ∈ Z+ and j ∈ {0, 1, 3, 4, 6, 7}.
1
1
1
1
1
1
1
1
1
K9m+3 = 3K3m+1 ∪ K3m+1,3m+1,3m+1
111
111
111
K3m+1 K3m+1 K3m+1
I.H.: K3m+1 dec. K3,K4,K6.
111
111
111
K3m+1,3m+1,3m+1
Thm 1: K3∗q dec. K3.
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Theorem 3: For r ≡ 0 or 1 (mod 3), Kr dec. K3,K4,K6
r = 9m + j for m ∈ Z+ and j ∈ {0, 1, 3, 4, 6, 7}.
1
1
1
1
1
1
1
1
1
K9m+3 = 3K3m+1 ∪ K3m+1,3m+1,3m+1
111
111
111
K3m+1 K3m+1 K3m+1
I.H.: K3m+1 dec. K3,K4,K6.
111
111
111
K3m+1,3m+1,3m+1
Thm 1: K3∗q dec. K3.
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Theorem 3: For r ≡ 0 or 1 (mod 3), Kr dec. K3,K4,K6
Example: r = 10.
1
1
1
1
1
1
1
1
1
1z
K10 = 3(K3 ∪ z) ∪ K3,3,3
1
1
1
1
1
1
1
1
1
1z
3(K3 ∪ z)Base Case: K4 dec. K3,K4,K6.
1
1
1
1
1
1
1
1
1
K3,3,3
Thm 1: K3∗q dec. K3.
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Theorem 3: For r ≡ 0 or 1 (mod 3), Kr dec. K3,K4,K6
Example: r = 10.
1
1
1
1
1
1
1
1
1
1z
K10 = 3(K3 ∪ z) ∪ K3,3,3
1
1
1
1
1
1
1
1
1
1z
3(K3 ∪ z)Base Case: K4 dec. K3,K4,K6.
1
1
1
1
1
1
1
1
1
K3,3,3
Thm 1: K3∗q dec. K3.
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Theorem 3: For r ≡ 0 or 1 (mod 3), Kr dec. K3,K4,K6
r = 9m + j for m ∈ Z+ and j ∈ {0, 1, 3, 4, 6, 7}.
1
1
1
1
1
1
1
1
1
1z
K9m+1 = 3(K3m ∪ z) ∪ K3m,3m,3m
111
111
111
1z
3(K3m ∪ z)I.H.: K3m+1 dec. K3,K4,K6.
111
111
111
K3m,3m,3m
Thm 1: K3∗q dec. K3.
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Theorem 3: For r ≡ 0 or 1 (mod 3), Kr dec. K3,K4,K6
r = 9m + j for m ∈ Z+ and j ∈ {0, 1, 3, 4, 6, 7}.
1
1
1
1
1
1
1
1
1
1z
K9m+1 = 3(K3m ∪ z) ∪ K3m,3m,3m
111
111
111
1z
3(K3m ∪ z)I.H.: K3m+1 dec. K3,K4,K6.
111
111
111
K3m,3m,3m
Thm 1: K3∗q dec. K3.
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Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Theorem 3: For r ≡ 0 or 1 (mod 3), Kr dec. K3,K4,K6
r = 9m + j for m ∈ Z+ and j ∈ {0, 1, 3, 4, 6, 7}. h ∈ {1, 3, 4}.
1
1
1
1
1
1
1
1
1
1
1
1
K9m+3+h =
3K3m+1 ∪ Kh ∪ K3m+1,3m+1,3m+1,h
111
111
111
111
K3m+1 K3m+1 K3m+1 Kh
I.H.: K3m+1 dec. K3,K4,K6.
111
111
111
111
K3m+1,3m+1,3m+1,h
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Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Theorem 3: For r ≡ 0 or 1 (mod 3), Kr dec. K3,K4,K6
r = 9m + j for m ∈ Z+ and j ∈ {0, 1, 3, 4, 6, 7}. h ∈ {1, 3, 4}.
1
1
1
1
1
1
1
1
1
1
1
1
K9m+3+h =
3K3m+1 ∪ Kh ∪ K3m+1,3m+1,3m+1,h
111
111
111
111
K3m+1 K3m+1 K3m+1 Kh
I.H.: K3m+1 dec. K3,K4,K6.
111
111
111
111
K3m+1,3m+1,3m+1,h
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Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Theorem 3: For r ≡ 0 or 1 (mod 3), Kr dec. K3,K4,K6
r = 9m + j for m ∈ Z+ and j ∈ {0, 1, 3, 4, 6, 7}. h ∈ {1, 3, 4}.
1
1
1
1
1
1
1
1
1
1
1
1
K9m+3+h =
3K3m+1 ∪ Kh ∪ K3m+1,3m+1,3m+1,h
111
111
111
111
K3m+1 K3m+1 K3m+1 Kh
I.H.: K3m+1 dec. K3,K4,K6.
111
111
111
111
K3m+1,3m+1,3m+1,3m+1
Thm 2: K4∗q dec. K4.
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Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Theorem 3: For r ≡ 0 or 1 (mod 3), Kr dec. K3,K4,K6
r = 9m + j for m ∈ Z+ and j ∈ {0, 1, 3, 4, 6, 7}. h ∈ {1, 3, 4}.
1
1
1
1
1
1
1
1
1
1
1
1
K9m+3+h =
3K3m+1 ∪ Kh ∪ K3m+1,3m+1,3m+1,h
111
111
111
111
K3m+1 K3m+1 K3m+1 Kh
I.H.: K3m+1 dec. K3,K4,K6.
111
111
111
111
K3m+1,3m+1,3m+1,h
K3m+1,3m+1,3m+1,h dec. K3,K4
(omit extra vertices of K4∗q dec. K4).
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Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Proof of Theorem 3: For r ≡ 0 or 1 (mod 3), Kr dec.K3,K4, and K6
Proof.
Suppose r = 9m. Partition V (K9m) into three sets A1,A2,A3
of size 3m.
Decompose K9m into 3 K3m induced by A1,A2,A3 and aK3∗3m with tripartition (A1,A2,A3).
By I.H., each K3m decomposes into a combo of K3,K4,K6.
K3∗3m decomposes into K3 by Thm 1.
Suppose r = 9m + 3. Partition V (K9m+3) into three sets ofsize 3m + 1.
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Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Proof of Theorem 3: For r ≡ 0 or 1 (mod 3), Kr dec.K3,K4, and K6
Proof.
Suppose r = 9m. Partition V (K9m) into three sets A1,A2,A3
of size 3m.
Decompose K9m into 3 K3m induced by A1,A2,A3 and aK3∗3m with tripartition (A1,A2,A3).
By I.H., each K3m decomposes into a combo of K3,K4,K6.
K3∗3m decomposes into K3 by Thm 1.
Suppose r = 9m + 3. Partition V (K9m+3) into three sets ofsize 3m + 1.
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Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Proof of Theorem 3: For r ≡ 0 or 1 (mod 3), Kr dec.K3,K4, and K6
Proof.
Suppose r = 9m. Partition V (K9m) into three sets A1,A2,A3
of size 3m.
Decompose K9m into 3 K3m induced by A1,A2,A3 and aK3∗3m with tripartition (A1,A2,A3).
By I.H., each K3m decomposes into a combo of K3,K4,K6.
K3∗3m decomposes into K3 by Thm 1.
Suppose r = 9m + 3. Partition V (K9m+3) into three sets ofsize 3m + 1.
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Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Proof of Theorem 3: For r ≡ 0 or 1 (mod 3), Kr dec.K3,K4, and K6
Proof.
Suppose r = 9m. Partition V (K9m) into three sets A1,A2,A3
of size 3m.
Decompose K9m into 3 K3m induced by A1,A2,A3 and aK3∗3m with tripartition (A1,A2,A3).
By I.H., each K3m decomposes into a combo of K3,K4,K6.
K3∗3m decomposes into K3 by Thm 1.
Suppose r = 9m + 3. Partition V (K9m+3) into three sets ofsize 3m + 1.
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Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Proof of Theorem 3: For r ≡ 0 or 1 (mod 3), Kr dec.K3,K4, and K6
Proof.
Suppose r = 9m. Partition V (K9m) into three sets A1,A2,A3
of size 3m.
Decompose K9m into 3 K3m induced by A1,A2,A3 and aK3∗3m with tripartition (A1,A2,A3).
By I.H., each K3m decomposes into a combo of K3,K4,K6.
K3∗3m decomposes into K3 by Thm 1.
Suppose r = 9m + 3. Partition V (K9m+3) into three sets ofsize 3m + 1.
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Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Proof of Theorem 3: For r ≡ 0 or 1 (mod 3), Kr dec.K3,K4, and K6
Proof.
Suppose r = 9m + 1. Partition V (K9m+1) into three sets ofsize 3m and a vertex z .
Decompose K9m+1 into 3 K3m+1 induced byA1 ∪ {z},A2 ∪ {z},A3 ∪ {z} and a K3∗3m with tripartition(A1,A2,A3).
By I.H., each K3m+1 decomposes into a combo of K3,K4,K6.
K3∗3m decomposes into K3 by Thm 1.
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Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Proof of Theorem 3: For r ≡ 0 or 1 (mod 3), Kr dec.K3,K4, and K6
Proof.
Suppose r = 9m + 1. Partition V (K9m+1) into three sets ofsize 3m and a vertex z .
Decompose K9m+1 into 3 K3m+1 induced byA1 ∪ {z},A2 ∪ {z},A3 ∪ {z} and a K3∗3m with tripartition(A1,A2,A3).
By I.H., each K3m+1 decomposes into a combo of K3,K4,K6.
K3∗3m decomposes into K3 by Thm 1.
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Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Proof of Theorem 3: For r ≡ 0 or 1 (mod 3), Kr dec.K3,K4, and K6
Proof.
Suppose r = 9m + 1. Partition V (K9m+1) into three sets ofsize 3m and a vertex z .
Decompose K9m+1 into 3 K3m+1 induced byA1 ∪ {z},A2 ∪ {z},A3 ∪ {z} and a K3∗3m with tripartition(A1,A2,A3).
By I.H., each K3m+1 decomposes into a combo of K3,K4,K6.
K3∗3m decomposes into K3 by Thm 1.
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Proof of Theorem 3: For r ≡ 0 or 1 (mod 3), Kr dec.K3,K4, and K6
Proof.
Suppose r = 9m + 1. Partition V (K9m+1) into three sets ofsize 3m and a vertex z .
Decompose K9m+1 into 3 K3m+1 induced byA1 ∪ {z},A2 ∪ {z},A3 ∪ {z} and a K3∗3m with tripartition(A1,A2,A3).
By I.H., each K3m+1 decomposes into a combo of K3,K4,K6.
K3∗3m decomposes into K3 by Thm 1.
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Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Proof of Theorem 3: For r ≡ 0 or 1 (mod 3), Kr dec.K3,K4, and K6
Proof.
Suppose r = 9m + 4, 9m + 6 or 9m + 7. PartitionV (K9m+3+h) into three sets of size 3m + 1 and a set of hvertices (h ∈ {1, 3, 4}).
Decompose K9m+3+h into 3 K3m+1 induced by A1,A2,A3, aK3m+1,3m+1,3m+1,h, and a K3 or K4 if h = 3 or h = 4 resp.
By I.H., each K3m+1 decomposes into a combo of K3,K4,K6.
K4∗3m+1 decomposes into K4 by Thm 2. This decomp.transforms into a decomp. of K3m+1,3m+1,3m+1,h into acombo of K3 and K4 by omitting the extra vertices (some K4
become K3).
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Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Proof of Theorem 3: For r ≡ 0 or 1 (mod 3), Kr dec.K3,K4, and K6
Proof.
Suppose r = 9m + 4, 9m + 6 or 9m + 7. PartitionV (K9m+3+h) into three sets of size 3m + 1 and a set of hvertices (h ∈ {1, 3, 4}).
Decompose K9m+3+h into 3 K3m+1 induced by A1,A2,A3, aK3m+1,3m+1,3m+1,h, and a K3 or K4 if h = 3 or h = 4 resp.
By I.H., each K3m+1 decomposes into a combo of K3,K4,K6.
K4∗3m+1 decomposes into K4 by Thm 2. This decomp.transforms into a decomp. of K3m+1,3m+1,3m+1,h into acombo of K3 and K4 by omitting the extra vertices (some K4
become K3).
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Proof of Theorem 3: For r ≡ 0 or 1 (mod 3), Kr dec.K3,K4, and K6
Proof.
Suppose r = 9m + 4, 9m + 6 or 9m + 7. PartitionV (K9m+3+h) into three sets of size 3m + 1 and a set of hvertices (h ∈ {1, 3, 4}).
Decompose K9m+3+h into 3 K3m+1 induced by A1,A2,A3, aK3m+1,3m+1,3m+1,h, and a K3 or K4 if h = 3 or h = 4 resp.
By I.H., each K3m+1 decomposes into a combo of K3,K4,K6.
K4∗3m+1 decomposes into K4 by Thm 2. This decomp.transforms into a decomp. of K3m+1,3m+1,3m+1,h into acombo of K3 and K4 by omitting the extra vertices (some K4
become K3).
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Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Proof of Theorem 3: For r ≡ 0 or 1 (mod 3), Kr dec.K3,K4, and K6
Proof.
Suppose r = 9m + 4, 9m + 6 or 9m + 7. PartitionV (K9m+3+h) into three sets of size 3m + 1 and a set of hvertices (h ∈ {1, 3, 4}).
Decompose K9m+3+h into 3 K3m+1 induced by A1,A2,A3, aK3m+1,3m+1,3m+1,h, and a K3 or K4 if h = 3 or h = 4 resp.
By I.H., each K3m+1 decomposes into a combo of K3,K4,K6.
K4∗3m+1 decomposes into K4 by Thm 2. This decomp.transforms into a decomp. of K3m+1,3m+1,3m+1,h into acombo of K3 and K4 by omitting the extra vertices (some K4
become K3).
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Necessary conditions are Sufficient for Steiner TripleSystems: Decomposition of Kn into K3
Theorem (Kirkman, 1847)
An (n, 3, 1) design exists ⇔ n ≡ 1 or 3 (mod 6).
i.e. Kn can be decomposed into K3 ⇔ n ≡ 1 or 3 (mod 6).
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Necessary conditions are Sufficient for Steiner TripleSystems: Decomposition of Kn into K3
Theorem (Kirkman, 1847)
An (n, 3, 1) design exists ⇔ n ≡ 1 or 3 (mod 6).
Let r = n−12 . Then, r ≡ 0 or 1 (mod 3).
1 1 1
Kr
Thm 3: Kr dec. K3,K4,K6.
→1 1 11 1 1
Kr∗2Kr∗2 dec. K3∗2,K4∗2,K6∗2.
Lemma 1: K2m+1 dec. K3 ⇔ Km∗2 dec. K3.By Lemma 1: K7,K9,K13 dec. K3 ⇒ K3∗2,K4∗2,K6∗2 dec. K3 ⇒Kr∗2 dec. K3.K2r+1 = Kn dec. K3 by Lemma 1.
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Necessary conditions are Sufficient for Steiner TripleSystems: Decomposition of Kn into K3
Theorem (Kirkman, 1847)
An (n, 3, 1) design exists ⇔ n ≡ 1 or 3 (mod 6).
Let r = n−12 . Then, r ≡ 0 or 1 (mod 3).
1 1 1
Kr
Thm 3: Kr dec. K3,K4,K6.
→1 1 11 1 1
Kr∗2Kr∗2 dec. K3∗2,K4∗2,K6∗2.
Lemma 1: K2m+1 dec. K3 ⇔ Km∗2 dec. K3.By Lemma 1: K7,K9,K13 dec. K3 ⇒ K3∗2,K4∗2,K6∗2 dec. K3 ⇒Kr∗2 dec. K3.K2r+1 = Kn dec. K3 by Lemma 1.
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Necessary conditions are Sufficient for Steiner TripleSystems: Decomposition of Kn into K3
Theorem (Kirkman, 1847)
An (n, 3, 1) design exists ⇔ n ≡ 1 or 3 (mod 6).
Let r = n−12 . Then, r ≡ 0 or 1 (mod 3).
1 1 1
Kr
Thm 3: Kr dec. K3,K4,K6.
→1 1 11 1 1
Kr∗2Kr∗2 dec. K3∗2,K4∗2,K6∗2.
Lemma 1: K2m+1 dec. K3 ⇔ Km∗2 dec. K3.By Lemma 1: K7,K9,K13 dec. K3 ⇒ K3∗2,K4∗2,K6∗2 dec. K3 ⇒Kr∗2 dec. K3.K2r+1 = Kn dec. K3 by Lemma 1.
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Necessary conditions are Sufficient for Steiner TripleSystems: Decomposition of Kn into K3
Theorem (Kirkman, 1847)
An (n, 3, 1) design exists ⇔ n ≡ 1 or 3 (mod 6).
Let r = n−12 . Then, r ≡ 0 or 1 (mod 3).
1 1 1
Kr
Thm 3: Kr dec. K3,K4,K6.
→1 1 11 1 1
Kr∗2Kr∗2 dec. K3∗2,K4∗2,K6∗2.
Lemma 1: K2m+1 dec. K3 ⇔ Km∗2 dec. K3.
By Lemma 1: K7,K9,K13 dec. K3 ⇒ K3∗2,K4∗2,K6∗2 dec. K3 ⇒Kr∗2 dec. K3.K2r+1 = Kn dec. K3 by Lemma 1.
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Necessary conditions are Sufficient for Steiner TripleSystems: Decomposition of Kn into K3
Theorem (Kirkman, 1847)
An (n, 3, 1) design exists ⇔ n ≡ 1 or 3 (mod 6).
Let r = n−12 . Then, r ≡ 0 or 1 (mod 3).
1 1 1
Kr
Thm 3: Kr dec. K3,K4,K6.
→1 1 11 1 1
Kr∗2Kr∗2 dec. K3∗2,K4∗2,K6∗2.
Lemma 1: K2m+1 dec. K3 ⇔ Km∗2 dec. K3.By Lemma 1: K7,K9,K13 dec. K3 ⇒ K3∗2,K4∗2,K6∗2 dec. K3 ⇒Kr∗2 dec. K3.
K2r+1 = Kn dec. K3 by Lemma 1.
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Necessary conditions are Sufficient for Steiner TripleSystems: Decomposition of Kn into K3
Theorem (Kirkman, 1847)
An (n, 3, 1) design exists ⇔ n ≡ 1 or 3 (mod 6).
Let r = n−12 . Then, r ≡ 0 or 1 (mod 3).
1 1 1
Kr
Thm 3: Kr dec. K3,K4,K6.
→1 1 11 1 1
Kr∗2Kr∗2 dec. K3∗2,K4∗2,K6∗2.
Lemma 1: K2m+1 dec. K3 ⇔ Km∗2 dec. K3.By Lemma 1: K7,K9,K13 dec. K3 ⇒ K3∗2,K4∗2,K6∗2 dec. K3 ⇒Kr∗2 dec. K3.K2r+1 = Kn dec. K3 by Lemma 1.
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Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Proof for Steiner Triple Systems:Kn dec. K3 ⇔ n ≡ 1 or 3 (mod 6)
Proof.
Suppose r = n−12 . Then, r ≡ 0 or 1 (mod 3).
By Thm 3, there’s a decomposition D of Kr into a combo ofK3,K4,K6. Replace each vertex of Kr by an independent setof 2 vertices to obtain Kr∗2.
For i ∈ {3, 4, 6}, each Ki in D corresponds to a Ki∗2 in Kr∗2.
Already showed K7,K9 and K13 could be decomposed into K3
and so by Lemma 1, K3∗2,K4∗2, and K6∗2 can too. Hence,Kr∗2 can be decomposed into K3.
K2r+1 = Kn can be decomposed into K3 by Lemma 1.
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Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Proof for Steiner Triple Systems:Kn dec. K3 ⇔ n ≡ 1 or 3 (mod 6)
Proof.
Suppose r = n−12 . Then, r ≡ 0 or 1 (mod 3).
By Thm 3, there’s a decomposition D of Kr into a combo ofK3,K4,K6. Replace each vertex of Kr by an independent setof 2 vertices to obtain Kr∗2.
For i ∈ {3, 4, 6}, each Ki in D corresponds to a Ki∗2 in Kr∗2.
Already showed K7,K9 and K13 could be decomposed into K3
and so by Lemma 1, K3∗2,K4∗2, and K6∗2 can too. Hence,Kr∗2 can be decomposed into K3.
K2r+1 = Kn can be decomposed into K3 by Lemma 1.
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Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Proof for Steiner Triple Systems:Kn dec. K3 ⇔ n ≡ 1 or 3 (mod 6)
Proof.
Suppose r = n−12 . Then, r ≡ 0 or 1 (mod 3).
By Thm 3, there’s a decomposition D of Kr into a combo ofK3,K4,K6. Replace each vertex of Kr by an independent setof 2 vertices to obtain Kr∗2.
For i ∈ {3, 4, 6}, each Ki in D corresponds to a Ki∗2 in Kr∗2.
Already showed K7,K9 and K13 could be decomposed into K3
and so by Lemma 1, K3∗2,K4∗2, and K6∗2 can too. Hence,Kr∗2 can be decomposed into K3.
K2r+1 = Kn can be decomposed into K3 by Lemma 1.
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Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Proof for Steiner Triple Systems:Kn dec. K3 ⇔ n ≡ 1 or 3 (mod 6)
Proof.
Suppose r = n−12 . Then, r ≡ 0 or 1 (mod 3).
By Thm 3, there’s a decomposition D of Kr into a combo ofK3,K4,K6. Replace each vertex of Kr by an independent setof 2 vertices to obtain Kr∗2.
For i ∈ {3, 4, 6}, each Ki in D corresponds to a Ki∗2 in Kr∗2.
Already showed K7,K9 and K13 could be decomposed into K3
and so by Lemma 1, K3∗2,K4∗2, and K6∗2 can too. Hence,Kr∗2 can be decomposed into K3.
K2r+1 = Kn can be decomposed into K3 by Lemma 1.
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Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Proof for Steiner Triple Systems:Kn dec. K3 ⇔ n ≡ 1 or 3 (mod 6)
Proof.
Suppose r = n−12 . Then, r ≡ 0 or 1 (mod 3).
By Thm 3, there’s a decomposition D of Kr into a combo ofK3,K4,K6. Replace each vertex of Kr by an independent setof 2 vertices to obtain Kr∗2.
For i ∈ {3, 4, 6}, each Ki in D corresponds to a Ki∗2 in Kr∗2.
Already showed K7,K9 and K13 could be decomposed into K3
and so by Lemma 1, K3∗2,K4∗2, and K6∗2 can too. Hence,Kr∗2 can be decomposed into K3.
K2r+1 = Kn can be decomposed into K3 by Lemma 1.
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Recall t-designs
Definition
A t − (n, k, λ) design is a set of subsets of size k (blocks) of a setV of n vertices such that all subsets of size t belong to exactly λblocks.
Abbreviated to (n, k , λ) design when t = 2.
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Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Recall t-designs
Definition
A t − (n, k, λ) design is a set of subsets of size k (blocks) of a setV of n vertices such that all subsets of size t belong to exactly λblocks.
Abbreviated to (n, k , λ) design when t = 2.
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Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
What else do we know?
Theorem (Kirkman, 1847 (k = 3) and Hanani, 1961 (k = 4), 1972(k = 5))
For λ = 1, t = 2, and k ∈ {3, 4, 5}, an (n, k , 1) design exists ⇔n(n − 1) ≡ 0 (mod k(k − 1)) and n − 1 ≡ 0 (mod k − 1).
For λ = 1, t = 2 and k = 6, no (36, 6, 1) design (Tarry, 1901).
36 officers problem (Euler, 1782): possible to arrange 6 regimentsconsisting of 6 officers each of different rank in a 6× 6 square sothat no rank or regiment will be repeated in any row or column.
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Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
What else do we know?
Theorem (Kirkman, 1847 (k = 3) and Hanani, 1961 (k = 4), 1972(k = 5))
For λ = 1, t = 2, and k ∈ {3, 4, 5}, an (n, k , 1) design exists ⇔n(n − 1) ≡ 0 (mod k(k − 1)) and n − 1 ≡ 0 (mod k − 1).
For λ = 1, t = 2 and k = 6, no (36, 6, 1) design (Tarry, 1901).
36 officers problem (Euler, 1782): possible to arrange 6 regimentsconsisting of 6 officers each of different rank in a 6× 6 square sothat no rank or regiment will be repeated in any row or column.
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
What else do we know?
Theorem (Kirkman, 1847 (k = 3) and Hanani, 1961 (k = 4), 1972(k = 5))
For λ = 1, t = 2, and k ∈ {3, 4, 5}, an (n, k , 1) design exists ⇔n(n − 1) ≡ 0 (mod k(k − 1)) and n − 1 ≡ 0 (mod k − 1).
For λ = 1, t = 2 and k = 6, no (36, 6, 1) design (Tarry, 1901).
36 officers problem (Euler, 1782): possible to arrange 6 regimentsconsisting of 6 officers each of different rank in a 6× 6 square sothat no rank or regiment will be repeated in any row or column.
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Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
All designs exist . . . eventually
Theorem (Wilson, 1973 (t = 2) and Keevash, 2014 (all t))
For all k , λ, t, there exists n0 such that for all n ≥ n0 satisfying thenecessary conditions, there exists a t − (n, k , λ) design.
For k = 6, λ = 1, t = 2, n0 ≤ 801.
If n ∈ {16, 21, 36, 46} then there is no (n, 6, 1) design.
For 21 values of n we don’t know if an (n, 6, 1) design exists.
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
All designs exist . . . eventually
Theorem (Wilson, 1973 (t = 2) and Keevash, 2014 (all t))
For all k , λ, t, there exists n0 such that for all n ≥ n0 satisfying thenecessary conditions, there exists a t − (n, k , λ) design.
For k = 6, λ = 1, t = 2, n0 ≤ 801.
If n ∈ {16, 21, 36, 46} then there is no (n, 6, 1) design.
For 21 values of n we don’t know if an (n, 6, 1) design exists.
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
All designs exist . . . eventually
Theorem (Wilson, 1973 (t = 2) and Keevash, 2014 (all t))
For all k , λ, t, there exists n0 such that for all n ≥ n0 satisfying thenecessary conditions, there exists a t − (n, k , λ) design.
For k = 6, λ = 1, t = 2, n0 ≤ 801.
If n ∈ {16, 21, 36, 46} then there is no (n, 6, 1) design.
For 21 values of n we don’t know if an (n, 6, 1) design exists.
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
All designs exist . . . eventually
Theorem (Wilson, 1973 (t = 2) and Keevash, 2014 (all t))
For all k , λ, t, there exists n0 such that for all n ≥ n0 satisfying thenecessary conditions, there exists a t − (n, k , λ) design.
For k = 6, λ = 1, t = 2, n0 ≤ 801.
If n ∈ {16, 21, 36, 46} then there is no (n, 6, 1) design.
For 21 values of n we don’t know if an (n, 6, 1) design exists.
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Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
One of the few other cases fully solved
Theorem (Hanani, 1960)
A 3− (n, 4, 1) design exists ⇔ n ≡ 2 or 4 (mod 6).
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Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Affine plane ((4,2,1) design)
1 1
1
1
Definition
System of points and lines s.t.
Two distinct points belongto exactly one line.
Let p be a point and l aline not containing p.Exactly one line parallel tol intersects p.
There exist at least 3points that are notcollinear.
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Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Affine planes and (q2, q, 1) resolvable designs
If in a finite affine plane, a line contains q points, then:
All lines contain q points.
All points belong to q + 1 lines.
There are q2 points and q2 + q lines.
There are q + 1 sets of q parallel lines with each line in eachset passing through all the points in the plane.
An affine plane is equivalent to a (q2, q, 1) resolvable design.
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Affine planes and (q2, q, 1) resolvable designs
If in a finite affine plane, a line contains q points, then:
All lines contain q points.
All points belong to q + 1 lines.
There are q2 points and q2 + q lines.
There are q + 1 sets of q parallel lines with each line in eachset passing through all the points in the plane.
An affine plane is equivalent to a (q2, q, 1) resolvable design.
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Affine planes and (q2, q, 1) resolvable designs
If in a finite affine plane, a line contains q points, then:
All lines contain q points.
All points belong to q + 1 lines.
There are q2 points and q2 + q lines.
There are q + 1 sets of q parallel lines with each line in eachset passing through all the points in the plane.
An affine plane is equivalent to a (q2, q, 1) resolvable design.
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Affine planes and (q2, q, 1) resolvable designs
If in a finite affine plane, a line contains q points, then:
All lines contain q points.
All points belong to q + 1 lines.
There are q2 points and q2 + q lines.
There are q + 1 sets of q parallel lines with each line in eachset passing through all the points in the plane.
An affine plane is equivalent to a (q2, q, 1) resolvable design.
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Affine planes and (q2, q, 1) resolvable designs
If in a finite affine plane, a line contains q points, then:
All lines contain q points.
All points belong to q + 1 lines.
There are q2 points and q2 + q lines.
There are q + 1 sets of q parallel lines with each line in eachset passing through all the points in the plane.
An affine plane is equivalent to a (q2, q, 1) resolvable design.
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Affine planes and (q2, q, 1) resolvable designs
If in a finite affine plane, a line contains q points, then:
All lines contain q points.
All points belong to q + 1 lines.
There are q2 points and q2 + q lines.
There are q + 1 sets of q parallel lines with each line in eachset passing through all the points in the plane.
An affine plane is equivalent to a (q2, q, 1) resolvable design.
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Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Affine plane to Projective plane((4,2,1) design to (7,3,1) design)
1 1
1
1
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Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Affine plane to Projective plane((4,2,1) design to (7,3,1) design)
1 1
1
1
1∞g
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Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Affine plane to Projective plane((4,2,1) design to (7,3,1) design)
1 1
1
1
1∞g
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Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Affine plane to Projective plane((4,2,1) design to (7,3,1) design)
1 1
1
1
1
1 1
∞r
∞g ∞b
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Affine plane to Projective plane((4,2,1) design to (7,3,1) design)
1 1
1
1
1
1 1
∞r
∞g ∞b
Fano Plane
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Affine plane to Projective plane((9,3,1) design to (13,4,1) design)
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Affine plane to Projective plane((9,3,1) design to (13,4,1) design)
∞v
∞b
∞r
∞g
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Affine plane to Projective plane((9,3,1) design to (13,4,1) design)
∞v
∞b
∞r
∞g
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Affine plane to Projective plane((9,3,1) design to (13,4,1) design)
∞v
∞b
∞r
∞g
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Projective plane
Definition
System of points and lines s.t.
Two distinct points belong to exactly one line.
Two lines intersect in exactly one point.
There exist 4 points such that any 3 of them are not aligned.
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Projective plane
Definition
System of points and lines s.t.
Two distinct points belong to exactly one line.
Two lines intersect in exactly one point.
There exist 4 points such that any 3 of them are not aligned.
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Projective plane
Definition
System of points and lines s.t.
Two distinct points belong to exactly one line.
Two lines intersect in exactly one point.
There exist 4 points such that any 3 of them are not aligned.
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Projective plane
Definition
System of points and lines s.t.
Two distinct points belong to exactly one line.
Two lines intersect in exactly one point.
There exist 4 points such that any 3 of them are not aligned.
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Projective planes and (q2 + q + 1, q + 1, 1) designs
A projective plane is of order q if each line contains q + 1 points.It has the following properties:
All lines contain q + 1 points.
All points belong to q + 1 lines.
There are q2 + q + 1 points and q2 + q + 1 lines.
A projective plane of order q is equivalent to a(q2 + q + 1, q + 1, 1) design.
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Projective planes and (q2 + q + 1, q + 1, 1) designs
A projective plane is of order q if each line contains q + 1 points.It has the following properties:
All lines contain q + 1 points.
All points belong to q + 1 lines.
There are q2 + q + 1 points and q2 + q + 1 lines.
A projective plane of order q is equivalent to a(q2 + q + 1, q + 1, 1) design.
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Projective planes and (q2 + q + 1, q + 1, 1) designs
A projective plane is of order q if each line contains q + 1 points.It has the following properties:
All lines contain q + 1 points.
All points belong to q + 1 lines.
There are q2 + q + 1 points and q2 + q + 1 lines.
A projective plane of order q is equivalent to a(q2 + q + 1, q + 1, 1) design.
Fionn Mc Inerney JCALM 2017 Decomposition of Complete Graphs
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Projective planes and (q2 + q + 1, q + 1, 1) designs
A projective plane is of order q if each line contains q + 1 points.It has the following properties:
All lines contain q + 1 points.
All points belong to q + 1 lines.
There are q2 + q + 1 points and q2 + q + 1 lines.
A projective plane of order q is equivalent to a(q2 + q + 1, q + 1, 1) design.
Fionn Mc Inerney JCALM 2017 Decomposition of Complete Graphs
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Projective planes and (q2 + q + 1, q + 1, 1) designs
A projective plane is of order q if each line contains q + 1 points.It has the following properties:
All lines contain q + 1 points.
All points belong to q + 1 lines.
There are q2 + q + 1 points and q2 + q + 1 lines.
A projective plane of order q is equivalent to a(q2 + q + 1, q + 1, 1) design.
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Properties of Graphs obtained from Projective planes
The bipartite graph G (P) for a given projective plane P where onevertex class consists of the points of P and the other the lines of Phas the following properties:
Bipartite.
High regular degree: q + 1.
High girth: 6.
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Properties of Graphs obtained from Projective planes
The bipartite graph G (P) for a given projective plane P where onevertex class consists of the points of P and the other the lines of Phas the following properties:
Bipartite.
High regular degree: q + 1.
High girth: 6.
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Properties of Graphs obtained from Projective planes
The bipartite graph G (P) for a given projective plane P where onevertex class consists of the points of P and the other the lines of Phas the following properties:
Bipartite.
High regular degree: q + 1.
High girth: 6.
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Properties of Graphs obtained from Projective planes
The bipartite graph G (P) for a given projective plane P where onevertex class consists of the points of P and the other the lines of Phas the following properties:
Bipartite.
High regular degree: q + 1.
High girth: 6.
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Projective planes and (q2 + q + 1, q + 1, 1) designs theorem
Theorem
If q is a prime power, then Kq2+q+1 decomposes into Kq+1.
All the same problem:
Kq2+q+1 decomposes into Kq+1?
A (q2 + q + 1, q + 1, 1) design exists?
A projective plane of order q exists?
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Projective planes and (q2 + q + 1, q + 1, 1) designs theorem
Theorem
If q is a prime power, then Kq2+q+1 decomposes into Kq+1.
All the same problem:
Kq2+q+1 decomposes into Kq+1?
A (q2 + q + 1, q + 1, 1) design exists?
A projective plane of order q exists?
Fionn Mc Inerney JCALM 2017 Decomposition of Complete Graphs
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Projective planes and (q2 + q + 1, q + 1, 1) designs theorem
Theorem
If q is a prime power, then Kq2+q+1 decomposes into Kq+1.
All the same problem:
Kq2+q+1 decomposes into Kq+1?
A (q2 + q + 1, q + 1, 1) design exists?
A projective plane of order q exists?
Fionn Mc Inerney JCALM 2017 Decomposition of Complete Graphs
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Projective planes and (q2 + q + 1, q + 1, 1) designs theorem
Theorem
If q is a prime power, then Kq2+q+1 decomposes into Kq+1.
All the same problem:
Kq2+q+1 decomposes into Kq+1?
A (q2 + q + 1, q + 1, 1) design exists?
A projective plane of order q exists?
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Non-existence of some designs
Theorem (Bruck-Ryser, 1949)
If q is not a prime power, q is not the sum of two squares, andq ≡ 1 or 2 (mod 4), then there does not exist a (q2, q, 1) design.
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Existence of Projective Planes
Theorem (Lam, 1991)
No projective plane of order q = 10 exists.
Proven by heavy computer calculations.
q = 12 is the first case where we don’t know if it exists.
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Existence of Projective Planes
Theorem (Lam, 1991)
No projective plane of order q = 10 exists.
Proven by heavy computer calculations.
q = 12 is the first case where we don’t know if it exists.
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Existence of Projective Planes
Theorem (Lam, 1991)
No projective plane of order q = 10 exists.
Proven by heavy computer calculations.
q = 12 is the first case where we don’t know if it exists.
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Conclusion
We know about the existence of designs for large values of nthanks to Wilson and Keevash.
Still many values of n below the threshold of n0 where wedon’t know if the designs exist.
While we may know a lot of designs exist, we don’t know howto construct many of them.
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Conclusion
We know about the existence of designs for large values of nthanks to Wilson and Keevash.
Still many values of n below the threshold of n0 where wedon’t know if the designs exist.
While we may know a lot of designs exist, we don’t know howto construct many of them.
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Conclusion
We know about the existence of designs for large values of nthanks to Wilson and Keevash.
Still many values of n below the threshold of n0 where wedon’t know if the designs exist.
While we may know a lot of designs exist, we don’t know howto construct many of them.
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OutlineIntroduction to Designs: Definitions and Examples
Steiner Triple Systems: Definition and ExistenceOther t-designs, Affine and Projective planes, and Conclusion
Thank you!
Fionn Mc Inerney JCALM 2017 Decomposition of Complete Graphs