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From Ramsey Theory to arithmetic progressions and hypergraphs Dhruv Mubayi Department of Mathematics, Statistics and Computer Science University of Illinois Chicago October 1, 2011 Dhruv Mubayi From Ramsey Theory to arithmetic progressions

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From Ramsey Theory to arithmetic progressions and hypergraphs

Dhruv Mubayi Department of Mathematics, Statistics and Computer Science University of Illinois Chicago October 1, 2011

Dhruv Mubayi From Ramsey Theory to arithmetic progressions and hypergraphs

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  • D. J. Kleitman

Of three ordinary people, two must have the same sex

Dhruv Mubayi From Ramsey Theory to arithmetic progressions and hypergraphs

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Ramsey Theory – total disorder is impossible

In any collection of six people, either three of them mutually know each other, or three of them mutually do not know each other.

Dhruv Mubayi From Ramsey Theory to arithmetic progressions and hypergraphs

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Ramsey Theory – total disorder is impossible

In any collection of six people, either three of them mutually know each other, or three of them mutually do not know each other. Is it true for five people?

Dhruv Mubayi From Ramsey Theory to arithmetic progressions and hypergraphs

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Suppose we seek four mutual acquaintances or four mutual

  • nonacquaintances. How many people are needed?

Dhruv Mubayi From Ramsey Theory to arithmetic progressions and hypergraphs

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Suppose we seek four mutual acquaintances or four mutual

  • nonacquaintances. How many people are needed?

What about p mutual acquaintances?

Dhruv Mubayi From Ramsey Theory to arithmetic progressions and hypergraphs

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Suppose we seek four mutual acquaintances or four mutual

  • nonacquaintances. How many people are needed?

What about p mutual acquaintances? Definition The Ramsey number R(p, p) is the minimum number of people such that we must have either p mutual acquaintances or p mutual nonacquaintances

Dhruv Mubayi From Ramsey Theory to arithmetic progressions and hypergraphs

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Suppose we seek four mutual acquaintances or four mutual

  • nonacquaintances. How many people are needed?

What about p mutual acquaintances? Definition The Ramsey number R(p, p) is the minimum number of people such that we must have either p mutual acquaintances or p mutual nonacquaintances R(3, 3) = 6

Dhruv Mubayi From Ramsey Theory to arithmetic progressions and hypergraphs

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Suppose we seek four mutual acquaintances or four mutual

  • nonacquaintances. How many people are needed?

What about p mutual acquaintances? Definition The Ramsey number R(p, p) is the minimum number of people such that we must have either p mutual acquaintances or p mutual nonacquaintances R(3, 3) = 6 R(4, 4) = 18

Dhruv Mubayi From Ramsey Theory to arithmetic progressions and hypergraphs

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43 ≤ R(5, 5) ≤ 49 102 ≤ R(6, 6) ≤ 165

Dhruv Mubayi From Ramsey Theory to arithmetic progressions and hypergraphs

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43 ≤ R(5, 5) ≤ 49 102 ≤ R(6, 6) ≤ 165 How many possible situations with 49 people?

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43 ≤ R(5, 5) ≤ 49 102 ≤ R(6, 6) ≤ 165 How many possible situations with 49 people? 2(49

2) = 21176 Dhruv Mubayi From Ramsey Theory to arithmetic progressions and hypergraphs

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Ramsey’s Theorem (finite case) R(p, p) is finite for every positive integer p. Moreover, ( √ 2)p < R(p, p) < 4p

Dhruv Mubayi From Ramsey Theory to arithmetic progressions and hypergraphs

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Ramsey’s Theorem (finite case) R(p, p) is finite for every positive integer p. Moreover, ( √ 2)p < R(p, p) < 4p No major improvements since the 1940’s

Dhruv Mubayi From Ramsey Theory to arithmetic progressions and hypergraphs

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Arithmetic Progressions (AP’s)

a a + d a + 2d a + 3d . . .

Dhruv Mubayi From Ramsey Theory to arithmetic progressions and hypergraphs

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Arithmetic Progressions (AP’s)

a a + d a + 2d a + 3d . . . 5 7 9 11 . . .

Dhruv Mubayi From Ramsey Theory to arithmetic progressions and hypergraphs

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Arithmetic Progressions (AP’s)

a a + d a + 2d a + 3d . . . 5 7 9 11 . . . 3 7 11 15

Dhruv Mubayi From Ramsey Theory to arithmetic progressions and hypergraphs

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Suppose we color the numbers 1, 2, 3 with red or blue. We are guaranteed an AP of length 2 in the same color.

Dhruv Mubayi From Ramsey Theory to arithmetic progressions and hypergraphs

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Suppose we color the numbers 1, 2, 3 with red or blue. We are guaranteed an AP of length 2 in the same color. What if we want an AP of length 3?

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Suppose we color the numbers 1, 2, 3 with red or blue. We are guaranteed an AP of length 2 in the same color. What if we want an AP of length 3? 9 numbers suffice but 8 do not! 1 2 3 4 5 6 7 8 What if we want an AP of length p?

Dhruv Mubayi From Ramsey Theory to arithmetic progressions and hypergraphs

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Definition W (p) is the minimum n such that every red-blue coloring of {1, 2, . . . , n} must contain a monochromatic AP of length p.

Dhruv Mubayi From Ramsey Theory to arithmetic progressions and hypergraphs

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Definition W (p) is the minimum n such that every red-blue coloring of {1, 2, . . . , n} must contain a monochromatic AP of length p. Van-der-Waerden’s Theorem (1927) W (p) is finite for every p.

Dhruv Mubayi From Ramsey Theory to arithmetic progressions and hypergraphs

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How big is W (p)?

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How big is W (p)?

W (2) = 3

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How big is W (p)?

W (2) = 3 W (3) = 9

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How big is W (p)?

W (2) = 3 W (3) = 9 W (4) = 35

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How big is W (p)?

W (2) = 3 W (3) = 9 W (4) = 35 W (5) = 178

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How big is W (p)?

W (2) = 3 W (3) = 9 W (4) = 35 W (5) = 178 W (6) = 1132

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How big is W (p)?

W (2) = 3 W (3) = 9 W (4) = 35 W (5) = 178 W (6) = 1132 W (p) < ???

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Big Functions

f1(x) = DOUBLE(x) = 2x

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Big Functions

f1(x) = DOUBLE(x) = 2x f2(x) = EXP(x) = 2x

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Big Functions

f1(x) = DOUBLE(x) = 2x f2(x) = EXP(x) = 2x EXP is obtained by applying DOUBLE x times starting at 1: 2x = f2(x) = 2 · 2 · 2 · · · 2 · 1 = f1(f1(f1(· · · f1(f1(1))))) where we iterate x times.

Dhruv Mubayi From Ramsey Theory to arithmetic progressions and hypergraphs

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f3(x) = TOWER(x)

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f3(x) = TOWER(x) TOWER(5) = f3(5) = 22222 = 265536

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f3(x) = TOWER(x) TOWER(5) = f3(5) = 22222 = 265536 f4(x) = WOW (x)

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f3(x) = TOWER(x) TOWER(5) = f3(5) = 22222 = 265536 f4(x) = WOW (x) For example, f4(4) is a tower of twos of height 65536.

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f3(x) = TOWER(x) TOWER(5) = f3(5) = 22222 = 265536 f4(x) = WOW (x) For example, f4(4) is a tower of twos of height 65536. Ackerman Function: g(x) = fx(x)

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f3(x) = TOWER(x) TOWER(5) = f3(5) = 22222 = 265536 f4(x) = WOW (x) For example, f4(4) is a tower of twos of height 65536. Ackerman Function: g(x) = fx(x) Ramsey’s Theorem implies that W (p) < g(p).

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f3(x) = TOWER(x) TOWER(5) = f3(5) = 22222 = 265536 f4(x) = WOW (x) For example, f4(4) is a tower of twos of height 65536. Ackerman Function: g(x) = fx(x) Ramsey’s Theorem implies that W (p) < g(p). Shelah’s Theorem W (p) < f4(5p)

Dhruv Mubayi From Ramsey Theory to arithmetic progressions and hypergraphs

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Conjecture W (p) < TOWER(p) for every p.

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Conjecture W (p) < TOWER(p) for every p. Ron Graham offered $1000 for this conjecture, and it was claimed by Tim Gowers in 1998 who proved that W (p) < 22222p+9

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Conjecture W (p) < TOWER(p) for every p. Ron Graham offered $1000 for this conjecture, and it was claimed by Tim Gowers in 1998 who proved that W (p) < 22222p+9 Now Graham offers $1000 for showing that W (p) < 2p2.

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Conjecture W (p) < TOWER(p) for every p. Ron Graham offered $1000 for this conjecture, and it was claimed by Tim Gowers in 1998 who proved that W (p) < 22222p+9 Now Graham offers $1000 for showing that W (p) < 2p2. Lower Bound W (p) > 2p

Dhruv Mubayi From Ramsey Theory to arithmetic progressions and hypergraphs

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Density Results

Erd˝

  • s-Tur´

an Conjecture Fix k ≥ 2 and ǫ > 0. Then for n sufficiently large, every subset S

  • f {1, 2, . . . , n} with |S| > ǫn contains a k-term AP.

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Density Results

Erd˝

  • s-Tur´

an Conjecture Fix k ≥ 2 and ǫ > 0. Then for n sufficiently large, every subset S

  • f {1, 2, . . . , n} with |S| > ǫn contains a k-term AP.

Szemer´ edi’s Theorem The Erd˝

  • s-Tur´

an Conjecture is true.

Dhruv Mubayi From Ramsey Theory to arithmetic progressions and hypergraphs

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Density Results

Erd˝

  • s-Tur´

an Conjecture Fix k ≥ 2 and ǫ > 0. Then for n sufficiently large, every subset S

  • f {1, 2, . . . , n} with |S| > ǫn contains a k-term AP.

Szemer´ edi’s Theorem The Erd˝

  • s-Tur´

an Conjecture is true. How large is “sufficiently large” ??

Dhruv Mubayi From Ramsey Theory to arithmetic progressions and hypergraphs

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Higher Dimensional Szemer´ edi Theorem

Multidimensional Szemer´ edi Theorem (Furstenberg-Katznelson) For every ǫ > 0, every positive integer r and every finite subset X ⊂ Zr there is a positive integer n such that every subset S of the grid {1, 2, . . . , n}r with |S| > ǫnr has a subset of the form

  • a + dX for some positive integer d.

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Higher Dimensional Szemer´ edi Theorem

Multidimensional Szemer´ edi Theorem (Furstenberg-Katznelson) For every ǫ > 0, every positive integer r and every finite subset X ⊂ Zr there is a positive integer n such that every subset S of the grid {1, 2, . . . , n}r with |S| > ǫnr has a subset of the form

  • a + dX for some positive integer d.

The Furstenberg-Katznelson proof gave no actual bound on n.

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Hypergraphs

Definition A k-uniform hypergraph on [n] := {1, 2, . . . , n} is a collection of k-element subsets of [n].

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Hypergraphs

Definition A k-uniform hypergraph on [n] := {1, 2, . . . , n} is a collection of k-element subsets of [n]. Definition A k-simplex is the k-uniform hypergraph on [k + 1] which consists

  • f all possible k-element sets (there are

k+1

k

  • = k + 1 of them).

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Graph Removal Lemmas

Question Suppose we have a graph ( = 2-uniform hypergraph) with few triangles ( = 2-simplices). Can we delete few edges so that after removing the edges, there are no triangles?

Dhruv Mubayi From Ramsey Theory to arithmetic progressions and hypergraphs

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Graph Removal Lemmas

Question Suppose we have a graph ( = 2-uniform hypergraph) with few triangles ( = 2-simplices). Can we delete few edges so that after removing the edges, there are no triangles? Rusza-Szemer´ edi (6,3) theorem For every a > 0 there exists c > 0 with the following property. If G is any graph with n vertices and at most cn3 triangles, then it is possible to remove at most an2 edges from G to make it triangle-free.

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Hypergraph Removal Lemma

Theorem (Frankl-R¨

  • dl, R¨
  • dl-Schacht, Gowers)

For every a > 0 there exists c > 0 with the following property. If H is any k-uniform hypergraph with n vertices and at most cnk+1 k-simplices, then it is possible to remove at most ank edges from H to make it k-simplex-free.

Dhruv Mubayi From Ramsey Theory to arithmetic progressions and hypergraphs

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Hypergraph Removal Lemma

Theorem (Frankl-R¨

  • dl, R¨
  • dl-Schacht, Gowers)

For every a > 0 there exists c > 0 with the following property. If H is any k-uniform hypergraph with n vertices and at most cnk+1 k-simplices, then it is possible to remove at most ank edges from H to make it k-simplex-free. A corollary to the removal lemma above is that we get an effective bound for n in the Furstenberg-Katznelson theorem.

Dhruv Mubayi From Ramsey Theory to arithmetic progressions and hypergraphs