chromatic ramsey number and circular chromatic ramsey number xuding zhu department of mathematics...
TRANSCRIPT
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Chromatic Ramsey Number andCircular Chromatic Ramsey Number
Xuding Zhu
Department of MathematicsZhejiang Normal University
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Among 6 people,
There are 3 know each other, or 3 do not know each other.
Know each other
Do not know each other
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Among 6 people,
There are 3 know each other, or 3 do not know each other.
![Page 4: Chromatic Ramsey Number and Circular Chromatic Ramsey Number Xuding Zhu Department of Mathematics Zhejiang Normal University](https://reader038.vdocuments.mx/reader038/viewer/2022102907/56649de85503460f94ae258e/html5/thumbnails/4.jpg)
Among 6 people,
There are 3 know each other, or 3 do not know each other.
![Page 5: Chromatic Ramsey Number and Circular Chromatic Ramsey Number Xuding Zhu Department of Mathematics Zhejiang Normal University](https://reader038.vdocuments.mx/reader038/viewer/2022102907/56649de85503460f94ae258e/html5/thumbnails/5.jpg)
Among 6 people,
There are 3 know each other, or 3 do not know each other.
![Page 6: Chromatic Ramsey Number and Circular Chromatic Ramsey Number Xuding Zhu Department of Mathematics Zhejiang Normal University](https://reader038.vdocuments.mx/reader038/viewer/2022102907/56649de85503460f94ae258e/html5/thumbnails/6.jpg)
Among 6 people,
There are 3 know each other, or 3 do not know each other.
Colour the edges of by red or blue,there is either a red or a blue
6K3K 3K
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Theorem [Ramsey] For any graphs G and H, there exists a graph F such that if the edges of F are coloured by red and blue, then there is a red copy of G or a blue copy of H
For `any’ systems , there exists a system F such that if `elements’ of F are partitioned into k parts, then for some i, the ith part contains as a subsystem.
k)i (1 i G
iG
General Ramsey Type Theorem:
Sufficiently large or complicated
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“Complete disorder is impossible”
A sufficiently large scale (or complicated) system must contains an interesting sub-system.
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There are Ramsey type theorems in many branches of mathematics such as combinatorics, number theory, geometry, ergodic theory, topology, combinatorial geometry, set theory, and measure theory.
Ramsey Theory has a wide range of applications.
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If the k-tuples M are t-colored, then
Theorem [Ramsey, 1927]
,,, knt,|| with set aFor mMM
nMMM |'| ,'
all the k-tuples of M’ having the same color.
principle pigeonhole :1k
graph complete a of edges thecolouring :2k
m
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For any partition of integers into finitely many parts, one part contains arithematical progression of arbitrarylarge length.
Van der Waeden Theorem
Szemerédi's theorem (1975)
Every set of integers A with positive density contains
arithematical progression of arbitrary length.
Timonthy Gowers [2001] gave a proof using both Fourier analysis and combinatorics.
Regularity lemmaErdos and Turan conjecture (1936)
Harmonic analysis
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Ramsey number R(3,k)
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For any 2-colouring of the edges of F with colours red and blue,there is a red copy of G or a blue copy of H.
),(K 336 KK
),( HGF means.
),(K 335 KK
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),( :min HGKnR(G,H) n
),(),( lkRKKR lk
The Ramsey number of (G,H) is
),( :min ,3 3 kn KKKnk)R(
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1933, George Szekeres, Esther Klein, Paul Erdos
starting with a geometric problem, Szekeres re-discovered Ramsey theorem, and proved 2 ,3 kk)R(
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Erdos [1946]
3/2
lnk
k ,3 k)R(
Erdos [1961]
Graver-Yackel [1968]
k
kOk)R(
ln
lnln k ,3 2
2
lnk
k ,3 k)R(
Ajtai-Komlos-Szemeredi[1980]
k
kOk)R(
ln ,3
2
Kim [1995]
k
kk)R(
ln ,3
2
2 ,3 kk)R( Szekere [1933]
Many sophisticated probabilistic tools are developed
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George Szekere and Esther Klein marriedlived together for 70 year, died on the same day 2005.8.28, within one hour.
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),( :min lkn KKKnR(k,l)
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Bounds for R(k,l)
k l 3 4 5 6 7 8
3 6 9 14 18 23 28
4 18 25 36
41
49
61
58
84
5 43
49
58
87
80
143
101
216
6 102
165
113
298
132
495
7 205
540
217
1031
8 282
1870
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Bounds for R(k,l)
k l 3 4 5 6 7 8
3 6 9 14 18 23 28
4 18 25 36
41
49
61
58
84
5 43
49
58
87
80
143
101
216
6 102
165
113
298
132
495
7 205
540
217
1031
8 282
1870
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Bounds for R(k,l)
k l 3 4 5 6 7 8
3 6 9 14 18 23 28
4 18 25 36
41
49
61
58
84
5 43
49
58
87
80
143
101
216
6 102
165
113
298
132
495
7 205
540
217
1031
8 282
1870
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How to measure a system?
A sufficiently large scale (or complicated) system must contains an interesting sub-system.
What is large scale?
What is complicated?
How to measure a graph?
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),( : min
is ,( ofnumber Ramsey The
HGKnR(G,H)
H)G
n
),( |:)(| min
is ,( ofnumber Ramsey The
HGFFVR(G,H)
H)G
),( |:)(| min
is ,( ofnumber Ramsey Size The
HGFFE(G,H)R
H)G
E
),( |:)(| min
is ,( ofnumber Ramsey -degree-max The
HGFF(G,H)R
H)G
),( :)( min
is ,( ofnumber Ramsey chromatic The
HGFF(G,H)R
H)G
),( :)( inf
is ,( ofnumber Ramsey chromaticcircular The
HGFF(G,H)R
H)G
cc
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Chromatic number
Circular chromatic number
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1,...,1,0: kVf
)()(~ yfxfyx
G=(V,E): a graph
an integer:1k
3k
An k-colouring of G is
0
1
20
1
A 3-colouring of 5Csuch that
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The chromatic number of G is
colouring-k a has G :kmin )( G
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Vf :
)()( yfxf
G=(V,E): a graph
an integer:1k
k-colouring of G is
such that yx ~
An
1|)()(|1 kyfxf
a real number
A (circular)
1,...,1,0 k),0[ k
0
1
20.5
1.5
A 2.5-coloring
1r
r-colouring of G is
),0[ r
1|)()(|1 ryfxf
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The circular chromatic number of G is
)(Gc { r: G has a circular r-colouring }infmin
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f is k-colouring of G
Therefore for any graph G,
)()( GGc
f is a circular k-colouring of G
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0=r
3
1
24
x~y |f(x)-f(y)|_r ≥ 1
The distance between p, p’ in the circle is
|'| |,'| min|p'-p| r pprpp
f is a circular r-colouring if
0 r
pp’
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Basic relation between )(G and )(Gc
).()(1)( GGG c
Circular chromatic number of a graph is a refinementof its chromatic number.
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Graph coloring is a model for resource distribution
Circular graph coloring is a model for resource distributionof periodic nature.
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),( :)( min
is ,( ofnumber Ramsey chromatic The
HGFF(G,H)R
H)G
Introduced by Burr-Erdos-Lovasz in 1976
),( lkR),K(KR lk
),( , HGRH)(GR
),( GGR(G)R
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If F has chromatic number , then there is a2 edge colouring of F in which each monochromaticsubgraph has chromatic number n-1.
2)1( n
),( GGF for any n-chromatic G.
4n
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If F has chromatic number , then there is a2 edge colouring of F in which each monochromaticsubgraph has chromatic number n-1.
2)1( n
),( GGF for any n-chromatic G.
1)1()( then ,)( If :nObservatio 2 nGRnG
1)1()( and )(G with graph a is there
n,each For :1976] Lovasz,-Erdos-[Burr Conjecture2 nGRnG
Could be much larger
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1)1()( and )(G with graph a is there
n,each For :1976] Lovasz,-Erdos-[Burr Conjecture2 nGRnG
The conjecture is true for n=3,4 (Burr-Erdos-Lovasz, 1976)
The conjecture is true for n=5 (Zhu, 1992)
The conjecture is true (Zhu, 2011)
Attempts by Tardif, West, etc. on non-diagonal casesof chromatic Ramsey numbers of graphs.
There are some upper bounds on nGG )(:)(Rmin
No more other case of the conjecture were verified, until 2011
1)1()(:)(Rmin 2 nnGG
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)hom(Kn
Lovasz]-Erdos-[Burr Lemma
n G(G)R
For any 2 edge-colouring of Kn, there is a monochromaticgraph which is a homomorphic image of G.
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Graph homomorphism = edge preserving map
GH
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1)1()( and )(G with graph a is there
n,each For :1976] Lovasz,-Erdos-[Burr Conjecture2 nGRnG
To prove Burr-Erdos-Lovasz conjecture for n, we need to construct an n-chromatic graph G, so that any 2 edge colouring of has a monochromatic subgraph which is a homomorphic image of G.
1)1( 2nK
The construction of G is easy:
Take all 2 edge colourings of 1)1( 2nK
mccc ,,, 21
For each 2 edge colouring ci of , one of the monochromaticsubgraph, say Gi, , has chromatic number at least n.
1)1( 2nK
iG
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To prove this conjecture for n, we need to construct an n-chromatic graph G, so that any 2 edge colouring of has a monochromatic subgraph which is a homomorphic image of G.
1)1( 2nK
The construction of G is easy:
Take all 2 edge colourings of 1)1( 2nK
mccc ,,, 21
For each 2 edge colouring of , one of the monochromaticsubgraph, say Gi, , has chromatic number at least n.
1)1( 2nK
iGic
mGGGG 21
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HG
G
H
GxH
Projections are homomorphisms
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To prove this conjecture for n, we need to construct an n-chromatic graph G, so that any 2 edge colouring of has a monochromatic subgraph which is a homomorphic image of G.
1)1( 2nK
The construction of G is easy:
Take all 2 edge colourings of 1)1( 2nK
mccc ,,, 21
For each 2 edge colouring ci of , one of the monochromaticsubgraph, say Gi, , has chromatic number at least n.
1)1( 2nK
mGGGG 21
GGi of image chomomorphi a is Each
iG
?
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G
H
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)(),(min)(
:1966] i,[Hedetniem Conjecture
HGHG
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To prove this conjecture for n, we need to construct an n-chromatic graph G, so that any 2 edge colouring of has a monochromatic subgraph which is a homomorphic image of G.
1)1( 2nK
?
If Hedetniemi’s conjecture is true, then
Burr-Erdos-Lovasz conjecture is true.
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GG of setst independen offamily :)(
A k-colouring of G partition V(G) into k independent sets.
1,0)(: G
1)()(,
GXXv
X
)(
)(minGX
X)(G
integer linear programming
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GG of setst independen offamily :)(
A k-colouring of G partition V(G) into k independent sets.
1,0)(: G
1)()(,
GXXv
X
)(
)(minGX
X)(G
]1,0[
)(Gf
ofnumber chromatic fractional :)( GGf
linear programming
)()( GGf
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)(),(min)(
:1966] i,[Hedetniem Conjecture
HGHG
)(),(min)(
:2002] [Z, Conjecture
HGHG fff
Fractional Hedetniemi’s conjecture
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To prove this conjecture for n, we need to construct an n-chromatic graph G, so that any 2 edge colouring of has a monochromatic subgraph which is a homomorphic image of G.
1)1( 2nK
If Hedetniemi’s conjecture is true, then
Burr-Erdos-Lovasz conjecture is true.
Observation: If fractional Hedetniemi’s conjecture is true, then Burr-Erdos-Lovasz conjecture is true.
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To prove this conjecture for n, we need to construct an n-chromatic graph G, so that any 2 edge colouring of has a monochromatic subgraph which is a homomorphic image of G.
1)1( 2nK
The construction of G is easy:
Take all 2 edge colourings of 1)1( 2nK
mccc ,,, 21
For each 2 edge colouring ci of , one of the monochromaticsubgraph, say Gi, , has chromatic number at least n.
1)1( 2nK
mGGGG 21
GGi of image chomomorphi a is Each
iG fractional chromatic number > n-1
1)( nGf 1)()( nGG f
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)(),(min)(
:1966] i,[Hedetniem Conjecture
HGHG
)(),(min)(
:2002] [Z, Conjecture
HGHG fff
Fractional Hedetniemi’s conjecture
)(),(min)(
:2002] [Z, Conjecture
HGHG fff
Theorem [Huajun Zhang, 2011]
If both G and H are vertex transitive, then
Theorem [Z, 2011]
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GG of setst independen offamily :)(
A k-colouring of G partition V(G) into k independent sets.
1,0)(: G
1)()(,
GXXv
X
)(
)(minGX
X)(G
]1,0[
)(Gf
ofnumber chromatic fractional :)( GGf
linear programming
]1,0[)(: GV
dual problem
1)()(,
GXXv
v
)(
)(maxGVv
v)(Gf
ofnumber clique fractional :)( GGf
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The fractional chromatic number of G is obtained by solving a linear programming problem
The fractional clique number of G is obtained by solving its dual problem
)()( GG ff
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)(),(min)(
:2002] [Z, Conjecture
HGHG fff
Fractional Hedetniemi’s conjecture is true
Theorem [Z, 2010]
)(),(min)( HGHG fff
:sketch oofPr
)(),(min weight with total
of clique fractional aconstruct tosuffices
HG
HG
ff
)(),(min)( HGHG fff
)(),(min)( HGHG fff
)(),(min)( HGHG fff
Easy!
Difficult!
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)(),(min weight with total
of clique fractional aconstruct tosuffices
HG
HG
ff
Easy
Easy
Difficult
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G of clique fractional maximum a ,10 : ],[V(G)g
H of clique fractional maximum a ],1,0[)( : HVh
(H)(G),ωω
g(x)h(y)(x,y)
HGV
ffmax
as defined ],1,0[)( :
)(),(min weight with total
of clique fractional a is
HG
HG
ff
Uyxff HGyhxg
HGU
),(
)(),(max)()(
, of set t independen
Easy!
Difficult!
![Page 57: Chromatic Ramsey Number and Circular Chromatic Ramsey Number Xuding Zhu Department of Mathematics Zhejiang Normal University](https://reader038.vdocuments.mx/reader038/viewer/2022102907/56649de85503460f94ae258e/html5/thumbnails/57.jpg)
),( :)( inf
is ,( ofnumber Ramsey chromaticcircular The
HGFF(G,H)R
H)G
cc
G)(GR(G)Rcc
,
What is the relation between and ? (G)Rc c(G)
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Basic relation between )(G and )(Gc
).()(1)( GGG c
).()( GGc
)(Gc )(Gis a refinement of
)(G )(Gcis an approximation of
![Page 59: Chromatic Ramsey Number and Circular Chromatic Ramsey Number Xuding Zhu Department of Mathematics Zhejiang Normal University](https://reader038.vdocuments.mx/reader038/viewer/2022102907/56649de85503460f94ae258e/html5/thumbnails/59.jpg)
There are many periodical scheduling problems in computer sciences.
The reciprocal of is studied by computer scientists as efficiency of a certain scheduling method, in 1986.
).(Gc
Circular colouring is a good model for periodical scheduling problems
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zGGRzR ccc )(:)(inf)(
nGGRnR )(:)(min)(
1)1()( :1976] Lovasz,-Erdos-[Burr Conjecture 2 nnR Theorem [Zhu, 2011]
?)( zRc No conjecture yet!
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).()()( GGG cf
.1)()( GG c
.largey arbitraril becan )()( GG f
)1( )( then ,integeran is 2 If kkkRkc
Using fractional version of Hedetniemi’s conjecture,Jao-Tardif-West-Zhu proved in 2014
)12(2 )2( c
R )13(3 )( 3 KRc
?)13(3 )3( c
R
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2 rational real becan )( Gc
?)( of valuespossible theareWhat x
GR
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),( :)( inf GGFF(G)R cc
colouring-circular a has : inf rGr(G)c min
min ?
No !
[ Jao-Tardif-West-Zhu, 2014]
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4 )( then ,2
52 If zRz
c
6 )( 3 KRc
9 ),( 43 KKRc
5 ),(3
14 53 CCR
c
2
9 ),(4 73 CCR
c
Some other results by Jao-Tardif-West-Zhu, 2014
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