erd˝os-szekeres-type theorems for monotone paths and...
TRANSCRIPT
![Page 1: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/1.jpg)
History
Erdos-Szekeres-type theoremsfor monotone paths and convex bodies
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk
November 24, 2010
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 2: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/2.jpg)
History
History
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Theorem
(Erdos-Szekeres 1935) For any positive integer n, there exists aninteger ES(n), such that any set of at least ES(n) points in theplane such that no three are collinear contains n members inconvex position. Moreover
2n−2 + 1 ≤ ES(n) ≤(
2n − 4
n − 2
)
+ 1 = O(4n/√
n).
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 3: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/3.jpg)
History
History
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Theorem
(Erdos-Szekeres 1935) For any positive integer n, there exists aninteger ES(n), such that any set of at least ES(n) points in theplane such that no three are collinear contains n members inconvex position. Moreover
2n−2 + 1 ≤ ES(n) ≤(
2n − 4
n − 2
)
+ 1 = O(4n/√
n).
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 4: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/4.jpg)
History
(c) 4-cup.��
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(d) 4-cap.
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Theorem
(Erdos-Szekeres 1935) For any positive integers k and l , thereexists an integer f (k, l), such that any set of at least f (k, l) pointsin the plane such that no three are collinear contains either a k-cupor an l-cap. Moreover
f (k, l) =
(
k + l − 4
k − 2
)
+ 1
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 5: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/5.jpg)
History
Proof is very combinatorial. The only geometric fact used was thefollowing: Order the points from left to right {p1, ..., pN}transitive property: If (p1, p2, p3) is a cap (cup), and (p2, p3, p4)is a cap (cup), then p1, p2, p3, p4 is a 4-cap (4-cup).
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Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 6: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/6.jpg)
History
Proof is very combinatorial. The only geometric fact used was thefollowing: Order the points from left to right {p1, ..., pN}transitive property: If (p1, p2, p3) is a cap (cup), and (p2, p3, p4)is a cap (cup), then p1, p2, p3, p4 is a 4-cap (4-cup).
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Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 7: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/7.jpg)
History
Proof is very combinatorial. The only geometric fact used was thefollowing: Order the points from left to right {p1, ..., pN}transitive property: If (p1, p2, p3) is a cap (cup), and (p2, p3, p4)is a cap (cup), then p1, p2, p3, p4 is a 4-cap (4-cup).
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2p
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Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 8: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/8.jpg)
History
Proof is very combinatorial. The only geometric fact used was thefollowing: Order the points from left to right {p1, ..., pN}transitive property: If (p1, p2, p3) is a cap (cup), and (p2, p3, p4)is a cap (cup), then p1, p2, p3, p4 is a 4-cap (4-cup).
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2p
p3
1p
p4
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 9: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/9.jpg)
History
Proof is very combinatorial. The only geometric fact used was thefollowing: Order the points from left to right {p1, ..., pN}transitive property: If (p1, p2, p3) is a cap (cup), and (p2, p3, p4)is a cap (cup), then p1, p2, p3, p4 is a 4-cap (4-cup).
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2p
p3
1p
p4
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 10: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/10.jpg)
History
transitive property:
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Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 11: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/11.jpg)
History
transitive property:
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Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 12: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/12.jpg)
History
transitive property:
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2p p
5
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Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 13: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/13.jpg)
History
transitive property:
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2p p
5
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Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 14: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/14.jpg)
History
transitive property:
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5
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Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 15: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/15.jpg)
History
Generalizing to convex bodies
Definition
A family C of convex bodies (compact convex sets) in the plane issaid to be in convex position if none of its members is contained inthe convex hull of the union of the others. We say that C is ingeneral position if every three members are in convex position.
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 16: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/16.jpg)
History
Theorem
(Bisztriczky and Fejes Toth 1989) For any positive integer n, thereexists an integer D(n), such that every family of at least D(n)disjoint convex bodies in the plane in general position contains nmembers in convex position. Moreover
2n−2 + 1 ≤ D(n) ≤ 222n
.
They conjectured D(n) = ES(n).Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 17: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/17.jpg)
History
Theorem
(Bisztriczky and Fejes Toth 1989) For any positive integer n, thereexists an integer D(n), such that every family of at least D(n)disjoint convex bodies in the plane in general position contains nmembers in convex position. Moreover
2n−2 + 1 ≤ D(n) ≤ 222n
.
They conjectured D(n) = ES(n).Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 18: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/18.jpg)
History
D(n) was later improved:
Theorem
(Pach and Toth 1998) D(n) ≤(
2n−4n−2
)2+ 1 = O(16n)
Theorem
(Hubard, Montejano, Mora, S. 2010)D(n) ≤ (
(2n−5n−2
)
+ 1)(2n−4
n−2
)
+ 1
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 19: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/19.jpg)
History
Definition
We say that a family of convex bodies in the plane is noncrossing ifany two members share at most two boundary points.
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 20: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/20.jpg)
History
Theorem
(Pach and Toth 2000) For any positive integer n, there exists aninteger N(n), such that any family of at least N(n) noncrossingconvex bodies in the plane in general position contains n membersin convex position. Moreover
2n−2 + 1 ≤ N(n) ≤ 222n
.
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 21: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/21.jpg)
History
Theorem
(Pach and Toth 2000) For any positive integer n, there exists aninteger N(n), such that any family of at least N(n) noncrossingconvex bodies in the plane in general position contains n membersin convex position. Moreover
2n−2 + 1 ≤ N(n) ≤ 222n
.
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 22: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/22.jpg)
History
Note: We cannot drop the noncrossing assumption. Pach and Tothgave a construction of n pairwise crossing rectangles that whichare in general position, but no four of them are in convex position
1 23 4
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 23: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/23.jpg)
History
Note: We cannot drop the noncrossing assumption. Pach and Tothgave a construction of n pairwise crossing rectangles that whichare in general position, but no four of them are in convex position
1 23 4
p
p
1
2
p1
3p
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 24: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/24.jpg)
History
Note: We cannot drop the noncrossing assumption. Pach and Tothgave a construction of n pairwise crossing rectangles that whichare in general position, but no four of them are in convex position
1 23 4
p
p1
2
p1
p4
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 25: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/25.jpg)
History
N(n) was later improved
Theorem
(Hubard, Montejano, Mora, S. 2010)
N(n) ≤ 22n
Proof introduces order types for convex bodies. Our result:
Theorem
(Fox, Pach, Sudakov, S.)
2n−2 + 1 ≤ N(n) ≤ nn2= 2cn2 log n.
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 26: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/26.jpg)
History
Order types of convex bodies
Given an ordering on C, (Ci ,Cj ,Ck) (i < j < k) has a clockwise(counterclockwise) orientation if there exist distinct pointspi ∈ Ci , pj ∈ Cj , pk ∈ Ck such that they lie on the boundary ofconv(Ci ∪ Cj ∪ Ck) and appear there in clockwise(counterclockwise) order. For i < j < k
Cj
iC
Ck
iC
Cj
Ck
CjiC
Ck
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 27: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/27.jpg)
History
Order types of convex bodies
Given an ordering on C, (Ci ,Cj ,Ck) (i < j < k) has a clockwise(counterclockwise) orientation if there exist distinct pointspi ∈ Ci , pj ∈ Cj , pk ∈ Ck such that they lie on the boundary ofconv(Ci ∪ Cj ∪ Ck) and appear there in clockwise(counterclockwise) order. For i < j < k
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Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 28: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/28.jpg)
History
Theorem
(Hubard, Montejano, Mora, S. 2010) A family C of noncrossingconvex sets is in covex position if and only if there exists anordering on the member of C such that every triple has a clockwiseorientation.
Which implies N(n) ≤ 22cnby Ramsey Theory.
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 29: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/29.jpg)
History
3
4
5
6 1
2
Theorem
(Hubard, Montejano, Mora, S. 2010) A family C of noncrossingconvex sets is in covex position if and only if there exists anordering on the member of C such that every triple has a clockwiseorientation.
Which implies N(n) ≤ 22cnby Ramsey Theory.
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 30: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/30.jpg)
History
3
4
5
6 1
2
Theorem
(Hubard, Montejano, Mora, S. 2010) A family C of noncrossingconvex sets is in covex position if and only if there exists anordering on the member of C such that every triple has a clockwiseorientation.
Which implies N(n) ≤ 22cnby Ramsey Theory.
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 31: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/31.jpg)
History
3
4
5
6 1
2
Theorem
(Hubard, Montejano, Mora, S. 2010) A family C of noncrossingconvex sets is in covex position if and only if there exists anordering on the member of C such that every triple has a clockwiseorientation.
Which implies N(n) ≤ 22cnby Ramsey Theory.
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 32: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/32.jpg)
History
3
4
5
6 1
2
Theorem
(Hubard, Montejano, Mora, S. 2010) A family C of noncrossingconvex sets is in covex position if and only if there exists anordering on the member of C such that every triple has a clockwiseorientation.
Which implies N(n) ≤ 22cnby Ramsey Theory.
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 33: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/33.jpg)
History
3
4
5
6 1
2
Theorem
(Hubard, Montejano, Mora, S. 2010) A family C of noncrossingconvex sets is in covex position if and only if there exists anordering on the member of C such that every triple has a clockwiseorientation.
Which implies N(n) ≤ 22cnby Ramsey Theory.
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 34: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/34.jpg)
History
However clockwise and counter clockwise orientations definedabove does not satisfy the transitive property.
C
C
2
C1
C3
4
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 35: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/35.jpg)
History
However clockwise and counter clockwise orientations definedabove does not satisfy the transitive property.
C
C
2
C1
C3
4
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 36: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/36.jpg)
History
However clockwise and counter clockwise orientations definedabove does not satisfy the transitive property.
C
C
2
C1
C3
4
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 37: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/37.jpg)
History
Strong orientations
Order the member of C from ”left to right” according to their ”leftendpoint”.
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 38: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/38.jpg)
History
Strong orientations
Order the member of C from ”left to right” according to their ”leftendpoint”.
C
C
1
C2
C3
C4
C5
6
7
C
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 39: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/39.jpg)
History
Given the ordering as above, for i < j < k, (Ci ,Cj ,Ck) is said tohave a strong-clockwise (strong-counterclockwise) orientation ifthere exist points pj ∈ Cj , pk ∈ Ck such that, starting at the leftendpoint p∗
i of Ci , the triple (p∗
i , pj , pk) appears in clockwise(counterclockwise) order along the boundary of conv(Ci ∪Cj ∪Ck).
Cj
iC
Ck
iC
Cj
Ck
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 40: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/40.jpg)
History
Given the ordering as above, for i < j < k, (Ci ,Cj ,Ck) is said tohave a strong-clockwise (strong-counterclockwise) orientation ifthere exist points pj ∈ Cj , pk ∈ Ck such that, starting at the leftendpoint p∗
i of Ci , the triple (p∗
i , pj , pk) appears in clockwise(counterclockwise) order along the boundary of conv(Ci ∪Cj ∪Ck).
����
����
����
pi
Cj
iC
Ck
p
p
j
k
*
����
����
����pi iC
Cj
Ck
j
kp
p
*
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 41: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/41.jpg)
History
(Ci ,Cj ,Ck) has both strong orientations if it has both astrong-clockwise and a strong-counterclockwise orientation.
CjiC
Ck
Note that the following only has a strong clockwise orientation.
3C
C2
C1
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 42: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/42.jpg)
History
(Ci ,Cj ,Ck) has both strong orientations if it has both astrong-clockwise and a strong-counterclockwise orientation.
���
���
��������
���
���
������
������
pi
pj
pj
pkCj
iCCk*
Note that the following only has a strong clockwise orientation.
��������
����
��������
p1
p2
p3
3C
C2
C1
*
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 43: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/43.jpg)
History
Theorem
(Fox, Pach, Sudakov, S.) transitive property: If (C1,C2,C3) and(C2,C3,C4) have only a strong clockwise (strongcounterclockwise) orientation, then (C1,C3,C4) and (C1,C2,C4) must alsoonly have strong clockwise (counter clockwise) orientations.
3C
C2
C1
C4
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 44: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/44.jpg)
History
Theorem
(Fox, Pach, Sudakov, S.) transitive property: If (C1,C2,C3) and(C2,C3,C4) have only a strong clockwise (strongcounterclockwise) orientation, then (C1,C3,C4) and (C1,C2,C4) must alsoonly have strong clockwise (counter clockwise) orientations.
3C
C2
C1
C4
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 45: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/45.jpg)
History
Theorem
(Fox, Pach, Sudakov, S.) transitive property: If (C1,C2,C3) and(C2,C3,C4) have only a strong clockwise (strongcounterclockwise) orientation, then (C1,C3,C4) and (C1,C2,C4) must alsoonly have strong clockwise (counter clockwise) orientations.
3C
C2
C1
C4
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 46: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/46.jpg)
History
Theorem
(Fox, Pach, Sudakov, S.) transitive property: If (C1,C2,C3) and(C2,C3,C4) have only a strong clockwise (strongcounterclockwise) orientation, then (C1,C3,C4) and (C1,C2,C4) must alsoonly have strong clockwise (counter clockwise) orientations.
3C
C2
C1
C4
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 47: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/47.jpg)
History
Theorem
(Fox, Pach, Sudakov, S.) transitive property: If (C1,C2,C3) and(C2,C3,C4) have both-strong orientations, then (C1,C3,C4) and(C1,C2,C4) must also have both-strong-orientations.
C1
C2C3 C4
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 48: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/48.jpg)
History
Theorem
(Fox, Pach, Sudakov, S.) transitive property: If (C1,C2,C3) and(C2,C3,C4) have both-strong orientations, then (C1,C3,C4) and(C1,C2,C4) must also have both-strong-orientations.
C1
C2C3 C4
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 49: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/49.jpg)
History
Theorem
(Fox, Pach, Sudakov, S.) transitive property: If (C1,C2,C3) and(C2,C3,C4) have both-strong orientations, then (C1,C3,C4) and(C1,C2,C4) must also have both-strong-orientations.
C1
C2C3 C4
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 50: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/50.jpg)
History
Theorem
(Fox, Pach, Sudakov, S.) transitive property: If (C1,C2,C3) and(C2,C3,C4) have both-strong orientations, then (C1,C3,C4) and(C1,C2,C4) must also have both-strong-orientations.
C1
C2C3 C4
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 51: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/51.jpg)
History
Combinatorial encoding.
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 52: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/52.jpg)
History
Combinatorial encoding.
����
����
����
����
����
����
����
����
����
C
C
1
5
7
C
C2
C3
C4
C8
1 2 3 4 5
C6C9
6 987
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 53: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/53.jpg)
History
Combinatorial encoding.
����
����
����
����
����
����
����
����
����
C
C
1
5
7
C
C2
C3
C4
C8
1 2 3 4 5
C6C9
6 987
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 54: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/54.jpg)
History
Combinatorial encoding.
����
����
����
����
����
����
����
����
����
C
C
1
5
7
C
C2
C3
C4
C8
1 2 3 4 5
C6C9
6 987
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 55: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/55.jpg)
History
Combinatorial encoding.
����
����
����
����
����
����
����
����
����
C
C
1
5
7
C
C2
C3
C4
C8
1 2 3 4 5
C6C9
6 987
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 56: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/56.jpg)
History
Find bodies in convex position by looking for a path and applyingthe transitive property.
����
����
����
����
����
����
1 2 6543
C
C
1
C2C3
4
C5
6C
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 57: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/57.jpg)
History
Find bodies in convex position by looking for a path and applyingthe transitive property.
����
����
����
����
����
����
1 2 6543
C
C
1
C2C3
4
C5
6C
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 58: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/58.jpg)
History
Find bodies in convex position by looking for a path and applyingthe transitive property.
����
����
����
����
����
����
1 2 6543
C
C
1
C2C3
4
C5
6C
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 59: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/59.jpg)
History
Find bodies in convex position by looking for a path and applyingthe transitive property.
����
����
����
����
����
����
1 2 6543
C
C
1
C2C3
4
C5
6C
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 60: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/60.jpg)
History
Find bodies in convex position by looking for a path and applyingthe transitive property.
����
����
����
����
����
����
1 2 6543
C
C
1
C2C3
4
C5
6C
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 61: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/61.jpg)
History
Find bodies in convex position by looking for a path and applyingthe transitive property.
����
����
����
����
����
����
1 2 6543
C
C
1
C2C3
4
C5
6C
By the transitive property, every triple has a strong clockwiseorientation. Hence by the previous theorem, C1, ...,C6 is in convexposition.
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 62: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/62.jpg)
History
Finding monochromatic paths in ordered hypergraphs
For an ordered 3-uniform hypergraph H = ([N],E ), a monotone3-path of length n are edges(v1, v2, v3), (v2, v3, v4), (v3, v4, v5), ..., (vn−2, vn−1, vn).
��������
����
����
����
����
��������
v1 v2v3 v4 v5 vn........
In general, for an ordered k-uniform hypergraph H = ([N],E ), amonotone k-path of length n are edges(v1, v2, ..., vk), (v2, v3, ..., vk+1)...., (vn−k+1 , ..., vn).
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 63: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/63.jpg)
History
Finding monochromatic paths in ordered hypergraphs
For an ordered 3-uniform hypergraph H = ([N],E ), a monotone3-path of length n are edges(v1, v2, v3), (v2, v3, v4), (v3, v4, v5), ..., (vn−2, vn−1, vn).
��������
����
����
����
����
��������
v1 v2v3 v4 v5 vn........
In general, for an ordered k-uniform hypergraph H = ([N],E ), amonotone k-path of length n are edges(v1, v2, ..., vk), (v2, v3, ..., vk+1)...., (vn−k+1 , ..., vn).
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 64: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/64.jpg)
History
Finding monochromatic paths in ordered hypergraphs
For an ordered 3-uniform hypergraph H = ([N],E ), a monotone3-path of length n are edges(v1, v2, v3), (v2, v3, v4), (v3, v4, v5), ..., (vn−2, vn−1, vn).
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v1 v2v3 v4 v5 vn........
In general, for an ordered k-uniform hypergraph H = ([N],E ), amonotone k-path of length n are edges(v1, v2, ..., vk), (v2, v3, ..., vk+1)...., (vn−k+1 , ..., vn).
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 65: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/65.jpg)
History
Finding monochromatic paths in ordered hypergraphs
For an ordered 3-uniform hypergraph H = ([N],E ), a monotone3-path of length n are edges(v1, v2, v3), (v2, v3, v4), (v3, v4, v5), ..., (vn−2, vn−1, vn).
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v1 v2v3 v4 v5 vn........
In general, for an ordered k-uniform hypergraph H = ([N],E ), amonotone k-path of length n are edges(v1, v2, ..., vk), (v2, v3, ..., vk+1)...., (vn−k+1 , ..., vn).
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 66: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/66.jpg)
History
Finding monochromatic paths in ordered hypergraphs
For an ordered 3-uniform hypergraph H = ([N],E ), a monotone3-path of length n are edges(v1, v2, v3), (v2, v3, v4), (v3, v4, v5), ..., (vn−2, vn−1, vn).
��������
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v1 v2v3 v4 v5 vn........
In general, for an ordered k-uniform hypergraph H = ([N],E ), amonotone k-path of length n are edges(v1, v2, ..., vk), (v2, v3, ..., vk+1)...., (vn−k+1 , ..., vn).
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 67: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/67.jpg)
History
Finding monochromatic paths in ordered hypergraphs
For an ordered 3-uniform hypergraph H = ([N],E ), a monotone3-path of length n are edges(v1, v2, v3), (v2, v3, v4), (v3, v4, v5), ..., (vn−2, vn−1, vn).
��������
����
����
����
����
��������
v1 v2v3 v4 v5 vn........
In general, for an ordered k-uniform hypergraph H = ([N],E ), amonotone k-path of length n are edges(v1, v2, ..., vk), (v2, v3, ..., vk+1)...., (vn−k+1 , ..., vn).
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 68: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/68.jpg)
History
Ordered hypergraphs
Definition
Let Nk(q, n) denote the smallest integer N such that every qcoloring on the k-tuples of [N] contains a monochromatic path oflength n.
N2(q, n) = (n − 1)q + 1 by Dilworth’s theorem.
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Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 69: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/69.jpg)
History
Ordered hypergraphs
Definition
Let Nk(q, n) denote the smallest integer N such that every qcoloring on the k-tuples of [N] contains a monochromatic path oflength n.
N2(q, n) = (n − 1)q + 1 by Dilworth’s theorem.
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Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 70: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/70.jpg)
History
Ordered hypergraphs
Definition
Let Nk(q, n) denote the smallest integer N such that every qcoloring on the k-tuples of [N] contains a monochromatic path oflength n.
N2(q, n) = (n − 1)q + 1 by Dilworth’s theorem.
���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ����
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 71: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/71.jpg)
History
Ordered hypergraphs
Definition
Let Nk(q, n) denote the smallest integer N such that every qcoloring on the k-tuples of [N] contains a monochromatic path oflength n.
N2(q, n) = (n − 1)q + 1 by Dilworth’s theorem.
���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ����
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 72: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/72.jpg)
History
N3(2, n) =(2n−4
n−2
)
+ 1 by the Erdos-Szekeres cups-caps (red-blue)argument.
���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ����
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 73: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/73.jpg)
History
N3(2, n) =(2n−4
n−2
)
+ 1 by the Erdos-Szekeres cups-caps (red-blue)argument.
���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ����
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 74: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/74.jpg)
History
N3(2, n) =(
2n−4n−2
)
+ 1 by the Erdos-Szekeres cups-caps (red-blue)argument.
���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ����
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 75: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/75.jpg)
History
N3(2, n) =(
2n−4n−2
)
+ 1 by the Erdos-Szekeres cups-caps (red-blue)argument.
���� ���� ���� ���� ���� ���� ���� ����
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 76: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/76.jpg)
History
N3(2, n) =(2n−4
n−2
)
+ 1 by the Erdos-Szekeres cups-caps (red-blue)argument.
���� ���� ���� ���� ���� ���� ���� ����
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 77: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/77.jpg)
History
N3(2, n) =(2n−4
n−2
)
+ 1 by the Erdos-Szekeres cups-caps (red-blue)argument.
���� ���� ���� ���� ���� ���� ���� ����
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 78: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/78.jpg)
History
N3(2, n) =(
2n−4n−2
)
+ 1 by the Erdos-Szekeres cups-caps (red-blue)argument.
���� ���� ���� ���� ���� ���� ���� ����
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 79: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/79.jpg)
History
N3(2, n) =(
2n−4n−2
)
+ 1 by the Erdos-Szekeres cups-caps (red-blue)argument.
���� ���� ���� ���� ���� ���� ���� ����
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 80: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/80.jpg)
History
N3(2, n) =(2n−4
n−2
)
+ 1 by the Erdos-Szekeres cups-caps (red-blue)argument.
���� ���� ���� ���� ���� ���� ���� ����
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 81: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/81.jpg)
History
For more colors.
Theorem
(Fox, Pach, Sudakov, S.) For q ≥ 3, we have
2(n/q)q−1 ≤ N3(q, n) ≤ nnq−1,
Therefore for the noncrossing convex bodies problem:
N(n) ≤ N3(3, n) ≤ nn2= 2cn2 log n
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 82: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/82.jpg)
History
Proof of N3(q, n) ≤ N2(nq−1, n) ≤ nnq−1
:
1 Set N = N2(nq−1, n)
2 χ :([N]
3
)
→ [q] be q-coloring on the triples of [N].
3 Then define φ :([N]
2
)
→ [n]q−1 as follows. We color
(i , j) ∈(
[N]2
)
with color (a1, a2, ..., aq−1) where at denotes thelength of the longest t-colored 3-path ending with vertices i , j .
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1 2 3 4 5 6 N...........
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 83: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/83.jpg)
History
Proof of N3(q, n) ≤ N2(nq−1, n) ≤ nnq−1
:
1 Set N = N2(nq−1, n)
2 χ :([N]
3
)
→ [q] be q-coloring on the triples of [N].
3 Then define φ :([N]
2
)
→ [n]q−1 as follows. We color
(i , j) ∈([N]
2
)
with color (a1, a2, ..., aq−1) where at denotes thelength of the longest t-colored 3-path ending with vertices i , j .
������
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��������
����
������
������
����
������
������
��������
1 2 3 4 5 6 N...........
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 84: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/84.jpg)
History
Proof of N3(q, n) ≤ N2(nq−1, n) ≤ nnq−1
:
1 Set N = N2(nq−1, n)
2 χ :(
[N]3
)
→ [q] be q-coloring on the triples of [N].
3 Then define φ :(
[N]2
)
→ [n]q−1 as follows. We color
(i , j) ∈([N]
2
)
with color (a1, a2, ..., aq−1) where at denotes thelength of the longest t-colored 3-path ending with vertices i , j .
������
������
����
����
���
���
����
��������
��������
1 2 3 4 5 6 N...........
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 85: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/85.jpg)
History
Proof of N3(q, n) ≤ N2(nq−1, n) ≤ nnq−1
:
1 Set N = N2(nq−1, n)
2 χ :([N]
3
)
→ [q] be q-coloring on the triples of [N].
3 Then define φ :([N]
2
)
→ [n]q−1 as follows. We color
(i , j) ∈(
[N]2
)
with color (a1, a2, ..., aq−1) where at denotes thelength of the longest t-colored 3-path ending with vertices i , j .
������
������
��������
����
������
������
���
���
��������
������
������
1 2 3 4 5 6 N...........
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 86: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/86.jpg)
History
Proof of N3(q, n) ≤ N2(nq−1, n) ≤ nnq−1
:
1 Set N = N2(nq−1, n)
2 χ :(
[N]3
)
→ [q] be q-coloring on the triples of [N].
3 Then define φ :(
[N]2
)
→ [n]q−1 as follows. We color
(i , j) ∈([N]
2
)
with color (a1, a2, ..., aq−1) where at denotes thelength of the longest t-colored 3-path ending with vertices i , j .
����
��������������������������������������������������������������������������������������������
i j
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 87: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/87.jpg)
History
Proof of N3(q, n) ≤ N2(nq−1, n) ≤ nnq−1
:
1 Set N = N2(nq−1, n)
2 χ :(
[N]3
)
→ [q] be q-coloring on the triples of [N].
3 Then define φ :(
[N]2
)
→ [n]q−1 as follows. We color
(i , j) ∈([N]
2
)
with color (a1, a2, ..., aq−1) where at denotes thelength of the longest t-colored 3-path ending with vertices i , j .
����
����
���
���
����
���
�������������������������������������������������������������������������������������������
i j
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 88: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/88.jpg)
History
Proof of N3(q, n) ≤ N2(nq−1, n) ≤ nnq−1
:
1 Set N = N2(nq−1, n)
2 χ :(
[N]3
)
→ [q] be q-coloring on the triples of [N].
3 Then define φ :(
[N]2
)
→ [n]q−1 as follows. We color
(i , j) ∈([N]
2
)
with color (a1, a2, ..., aq−1) where at denotes thelength of the longest t-colored 3-path ending with vertices i , j .
����
����
���
���
����
���
���
���
���
����������������������������������������������������������������������������������������
i j
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 89: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/89.jpg)
History
Proof of N3(q, n) ≤ N2(nq−1, n) ≤ nnq−1
:
1 Set N = N2(nq−1, n)
2 χ :(
[N]3
)
→ [q] be q-coloring on the triples of [N].
3 Then define φ :([N]
2
)
→ [n]q−1 as follows. We color
(i , j) ∈([N]
2
)
with color (a1, a2, ..., aq−1) where at denotes thelength of the longest t-colored 3-path ending with vertices i , j .
����
����
����
���
���
����
����
����������������������������������������������������������������������������������������
i j
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 90: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/90.jpg)
History
Proof of N3(q, n) ≤ N2(nq−1, n) ≤ nnq−1
:
1 Set N = N2(nq−1, n)
2 χ :(
[N]3
)
→ [q] be q-coloring on the triples of [N].
3 Then define φ :([N]
2
)
→ [n]q−1 as follows. We color
(i , j) ∈([N]
2
)
with color (a1, a2, ..., aq−1) where at denotes thelength of the longest t-colored 3-path ending with vertices i , j .
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i j
6( ,
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 91: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/91.jpg)
History
Proof of N3(q, n) ≤ N2(nq−1, n) ≤ nnq−1
:
1 Set N = N2(nq−1, n)
2 χ :([N]
3
)
→ [q] be q-coloring on the triples of [N].
3 Then define φ :(
[N]2
)
→ [n]q−1 as follows. We color
(i , j) ∈([N]
2
)
with color (a1, a2, ..., aq−1) where at denotes thelength of the longest t-colored 3-path ending with vertices i , j .
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i j
6( ,
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 92: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/92.jpg)
History
Proof of N3(q, n) ≤ N2(nq−1, n) ≤ nnq−1
:
1 Set N = N2(nq−1, n)
2 χ :(
[N]3
)
→ [q] be q-coloring on the triples of [N].
3 Then define φ :(
[N]2
)
→ [n]q−1 as follows. We color
(i , j) ∈([N]
2
)
with color (a1, a2, ..., aq−1) where at denotes thelength of the longest t-colored 3-path ending with vertices i , j .
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���
����������������������������������������������������������������������������������������
i j
6( ,
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 93: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/93.jpg)
History
Proof of N3(q, n) ≤ N2(nq−1, n) ≤ nnq−1
:
1 Set N = N2(nq−1, n)
2 χ :([N]
3
)
→ [q] be q-coloring on the triples of [N].
3 Then define φ :(
[N]2
)
→ [n]q−1 as follows. We color
(i , j) ∈([N]
2
)
with color (a1, a2, ..., aq−1) where at denotes thelength of the longest t-colored 3-path ending with vertices i , j .
����
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����
��������������������������������������������������������������������������������������
i j
6( ,5,......
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 94: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/94.jpg)
History
By definition of N2(nq−1, n), there is monochromatic 2-path on
vertices v1 < v2 < ... < vn with color (a∗1, ..., a∗
q−1).
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v1v2
v3 vnv
n−1
,a*2(a1* ,...,aq−1* ) ,a*2(a1* ,...,aq−1* ) ,a*2(a1* ,...,aq−1* )
........
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 95: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/95.jpg)
History
Claim: (v1, ..., vn) is a monochromatic 3-path (with color q)!Indeed, Assume (vi , vi+1, vi+2) has color j 6= q.
1 Longest jth-colored 3-path ending with vertices (vi , vj ) mustbe shorter than the longest jth-colored 3-path ending withvertices (vj+1, vj+2).
2 Contradicts φ(vi , vi+1) = φ(vi+1, vi+2).
3 Hence (vi , vi+1, vi+2) must have color q for all i .
��������
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����
viv
i+2vi+1
1*(a q−1* )aaj*,.., ,..,
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 96: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/96.jpg)
History
Claim: (v1, ..., vn) is a monochromatic 3-path (with color q)!Indeed, Assume (vi , vi+1, vi+2) has color j 6= q.
1 Longest jth-colored 3-path ending with vertices (vi , vj ) mustbe shorter than the longest jth-colored 3-path ending withvertices (vj+1, vj+2).
2 Contradicts φ(vi , vi+1) = φ(vi+1, vi+2).
3 Hence (vi , vi+1, vi+2) must have color q for all i .
��������
��������
����
viv
i+2vi+1
1*(a q−1* )aaj*,.., ,..,
,.., q−1* )aaj*,..,1*(a +1
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 97: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/97.jpg)
History
The upper bound proof can easily be generalized to show
Nk(q, n) ≤ Nk−1((n − k + 1)q−1, n)
Using the stepping-up approach we have
Theorem
(Fox, Pach, Sudakov, S.) Define t1(x) = x and ti+1(x) = 2ti (x).Then for k ≥ 4 we have
tk−1(cnq−1) ≤ Nk(q, n) ≤ tk−1(c
′nq−1 log n).
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 98: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/98.jpg)
History
Another upper bound on N3(q, n)
Consider the following game played by two players, Builder andPainter.
1 vertex vt+1 is revealed.
2 Builder decides whether to draw the edge (vi , vt+1) for i ≤ t.item If Builder draws an edge, Painter must immediately colorit one of q colors.
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v1 v2 vt−1 vtvt+1
.....
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 99: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/99.jpg)
History
Another upper bound on N3(q, n)
Consider the following game played by two players, Builder andPainter.
1 vertex vt+1 is revealed.
2 Builder decides whether to draw the edge (vi , vt+1) for i ≤ t.item If Builder draws an edge, Painter must immediately colorit one of q colors.
������
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��������
����
����
������
������
v1 v2 vt−1 vtvt+1
.....
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 100: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/100.jpg)
History
Another upper bound on N3(q, n)
Consider the following game played by two players, Builder andPainter.
1 vertex vt+1 is revealed.
2 Builder decides whether to draw the edge (vi , vt+1) for i ≤ t.item If Builder draws an edge, Painter must immediately colorit one of q colors.
������
������
��������
����
����
������
������
v1 v2 vt−1 vtvt+1
.....
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 101: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/101.jpg)
History
Another upper bound on N3(q, n)
Consider the following game played by two players, Builder andPainter.
1 vertex vt+1 is revealed.
2 Builder decides whether to draw the edge (vi , vt+1) for i ≤ t.item If Builder draws an edge, Painter must immediately colorit one of q colors.
������
������
��������
����
����
������
������
v1 v2 vt−1 vtvt+1
.....
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 102: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/102.jpg)
History
Another upper bound on N3(q, n)
Consider the following game played by two players, Builder andPainter.
1 vertex vt+1 is revealed.
2 Builder decides whether to draw the edge (vi , vt+1) for i ≤ t.item If Builder draws an edge, Painter must immediately colorit one of q colors.
������
������
��������
����
����
������
������
v1 v2 vt−1 vtvt+1
.....
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 103: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/103.jpg)
History
The vertex online Ramsey number V2(q, n) is the minimumnumber of edges builder has to draw to guarantee amonochromatic path of length n. Clearly V2(q, n) ≤
((n−1)q+12
)
.
Theorem
(Fox, Pach, Sudakov, S.) We have
V2(q, n) ≤ q2nq log n
Theorem
(Fox, Pach, Sudakov, S.) We have
N3(q, n) ≤ qV2(q,n) + 1 = qq2nq log n
For q = 3, the formula above implies N3(3, n) ≤ 2cn3 log n (Not asstrong as the previous bound 2c′n2 log n). A weaker upper bound,but gives us an algorithm of finding a monochromatic 3-path oflength n.
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 104: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/104.jpg)
History
Summary
1 Points (Toth and Valtr 2005):
2n−1 + 1 ≤ ES(n) ≤(
2n − 5
n − 2
)
+ 1 = O(4n/√
n).
2 Disjoint convex bodies (Hubard, Montejano, Mora, S. 2010):
2n−1 + 1 ≤ D(n) ≤ (
(
2n − 5
n − 2
)
+ 1)
(
2n − 4
n − 2
)
+ 1 = O(16n).
3 Noncrossing convex bodies (Fox, Pach, Sudakov, S.):
2n−1 + 1 ≤ N(n) ≤ nn2= 2O(n2 log n).
Problem
ES(n) = D(n) = N(n)?
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b
![Page 105: Erd˝os-Szekeres-type theorems for monotone paths and ...homepages.math.uic.edu/~suk/bernoulli2.pdfHistory (c) 4-cup. (d) 4-cap. Theorem (Erdo˝s-Szekeres 1935) For any positive integers](https://reader034.vdocuments.mx/reader034/viewer/2022052104/603fd287f56257253971c2f2/html5/thumbnails/105.jpg)
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Thank you!
Jacob Fox, Janos Pach, Benny Sudakov, Andrew Suk Erdos-Szekeres-type theorems for monotone paths and convex b