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Transcript of Presented by: Ryan ODonnell Carnegie Melloni by D. H. J. Polymath

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presented by: Ryan ODonnell Carnegie Melloni by D. H. J. Polymath Slide 2 Slide 3 Slide 4 If A {0,1,2} n has density (1), then A contains a (combinatorial) line. DHJ(3): point:1 1 2 1 2 1 0 0 0 1 0 1 1 0 1 2 0 1 2 2 line: 1 0 2 1 1 2 1 1 0 0 0 1 0 0 0 1 1 0 1 1 1 0 2 1 1 2 1 1 0 1 1 1 0 0 1 1 1 1 1 1 1 0 2 1 1 2 1 1 0 2 2 1 0 0 2 1 1 2 1 1 { },, 1 0 2 1 1 2 1 1 0 1 0 0 1 1 1 1 Slide 5 [Furstenberg-Katznelson91] : DHJ(k) is true k. (k = 3): > 0, n 0 () s.t. n n 0 (), A {0,1,2} n has density A contains a line proof: Used ergodic theory. No effective bound on n 0 (). Slide 6 Importance in combinatorics: Implies Szemerdis Theorem. 1 0 2 1 1 2 1 1 0 0 0 1 0 0 0 1 1 0 1 1 1 0 2 1 1 2 1 1 0 1 1 1 0 0 1 1 1 1 1 1 1 0 2 1 1 2 1 1 0 2 2 1 0 0 2 1 1 2 1 1 Slide 7 Why I was interested: 0 1 2 0 0, 1, 2, 1 0 1 2 2 A {0,1,2} n, no trips w/ all cols 0 0 0 1 1 0, 0, 1, 0, 1 0 1 0 0 1 A {0,1} n, no trips w/ all cols see [O-Wu09] Slide 8 Slide 9 Slide 10 Slide 11 Slide 12 Slide 13 Slide 14 Slide 15 Slide 16 Slide 17 Slide 18 Slide 19 Slide 20 Slide 21 [D.H.J. Polymath]: > 0, n 2 O(1/ 3 ), A {0,1,2} n has density A contains a line proof: Elementary probability/combinatorics. also the general k case, with Ackermann-type bounds Slide 22 Proof sketch Slide 23 DHJ(2): A {0,1} n dens. A contains line line: 1 0 1 1 1 1 1 1 0 0 0 1 0 0 0 1 1 0 1 1 1 0 1 1 1 1 1 1 0 1 1 1 0 0 1 1 1 1 1 1 { }, 1 0 1 1 1 1 1 1 0 1 0 0 1 1 1 1 contrapositive: If A has no comparable pairs i.e., A is an antichain then A is very sparse. Sperners Theorem Slide 24 DHJ(2): A {0,1} n dens. A contains line 1234567n 00011111 equal-slices distribution Slide 25 DHJ(2): A {0,1} n dens. A contains line equal-slices distribution 5 0 n 0 2 0 4 1 1 1 3 1 6 1 7 1 1 1 7 1 6 1 3 1 4 1 2 0 n 0 5 0 (each Hamming wt. equally likely) Slide 26 DHJ(2): A {0,1} n dens. A contains line 5 0 n 0 2 0 4 1 1 1 3 1 6 1 7 1 1 0 1 1 0 1 1 0 1234567n equal-slices distribution DHJ(2): A {0,1} n eq-slices dens. A contains line Slide 27 5 0 n 0 2 0 4 1 1 1 3 1 6 1 7 1 1 0 1 1 0 1 1 0 1234567n DHJ(2): A {0,1} n eq-slices dens. A contains line 00000011 Slide 28 5 0 n 0 2 0 4 1 1 1 3 1 6 1 7 1 1 0 1 1 0 1 1 0 1234567n DHJ(2): A {0,1} n eq-slices dens. A contains line 0000001100000011 0 0 0 0 0 1 1 0 0000001100000011 Slide 29 5 0 n 0 2 0 4 1 1 1 3 1 6 1 7 1 1 0 1 1 0 1 1 0 1234567n DHJ(2): A {0,1} n eq-slices dens. A contains line 0000001100000011 0 0 0 0 0 1 1 0 00000011 Slide 30 5 0 n 0 2 0 4 1 1 1 3 1 6 1 7 1 1 0 1 1 0 1 1 0 DHJ(2): A {0,1} n eq-slices dens. A contains line 0000001100000011 0 0 0 0 0 1 1 0 00000011 ( Pr[degen] = ) random eq-slices line Slide 31 DHJ(2): A {0,1} n eq-slices dens. eq-slice-line in A w.p. 2 Pr[ x, y A ] = E xy x y A A & Pr = E x x A 2 Pr E x x A 2 = 2 (independent) Slide 32 DHJ(2): A {0,1} n eq-slices dens. eq-slice-line in A w.p. 2 Slide 33 DHJ(2): A {0,1} n eq-slices dens. n 1/ 2 A contains a line Slide 34 DHJ(2): A {0,1} n eq-slices dens. Distribution Hackery 1 0 1 1 1 1 1 1 0 0 0 1 0 0 0 1 1 0 1 1unif rand. coords unif on eq-slices on 1 0 0 1 1 0 1 1 1 1 1 0 0 0 1 0 0 1 0 1 Slide 35 [ eq-slices ( A ) ] DHJ(2): A {0,1} n eq-slices dens. Distribution Hackery 1 0 1 1 1 1 1 1 0 0 0 1 0 0 0 1 1 0 1 1unif 1 0 0 1 1 0 1 1 1 1 1 0 0 0 1 0 0 1 0 1 easy: d TV ( unif, unif ) = o(1)unif (A) = E [ eq-slices A y y rand. coords Slide 36 5 0 n 0 2 0 4 1 1 1 3 1 6 1 7 1 1 0 1 1 0 1 1 0 0000001100000011 0 0 0 0 0 1 1 0 00000011 1234567n DHJ(2): A {0,1} n eq-slices dens. eq-slice-line in A w.p. 2 Slide 37 DHJ(3): A {0,1,2} n eq-slices dens. eq-slice-line in A w.p. 3 ??? Slide 38 Pr[ x, y A ] = E xy x y A A & Pr = E x y x A 2 Pr E xy x A 2 = 2 (independent) Slide 39 5 0 n 0 2 0 4 1 1 1 3 1 6 1 7 1 1 0 1 1 0 1 1 0 0 0 0 0 0 1 1 0 DHJ(3): A {0,1,2} n eq-slices dens. ? 00000011 1234567n Slide 40 5 0 n 0 2 0 4 1 1 1 3 1 6 1 7 1 1 0 1 1 0 1 1 0 00000011 0 0 0 0 0 1 1 0 1234567n DHJ(3): A {0,1,2} n eq-slices dens. 00022211 2 0 2 2 0 1 1 0 eq-slices pt in {0,1,2} n eq-slices line over {0,1} n if in A as a line, & in A as a point, done! Slide 41 DHJ(3): A {0,1,2} n eq-slices dens. by distrib. hackery: A {0,1} n eq-slices dens. also {0,1,2} n {0,1} n A A Slide 42 DHJ(3): A {0,1,2} n eq-slices dens. by distrib. hackery: A {0,1} n eq-slices dens. also {0,1,2} n {0,1} n A A L L = { x {0,1,2} n : x 21, x 20 A } 2 Slide 43 L = { x {0,1,2} n : x 21, x 20 A } {0,1,2} n {0,1} n A A L 2 L = { x {0,1,2} n : x 21, x 20 A } {0,1,2} n {0,1} n A A L 2 Slide 44 L = { x {0,1,2} n : x 21, x 20 A } {0,1,2} n {0,1} n A A L 2 if L A, done! Slide 45 L = { x {0,1,2} n : x 21, x 20 A } {0,1,2} n {0,1} n A A L 2 idea: A has dens. in Slide 46 L = { x {0,1,2} n : x 21, x 20 A } {0,1,2} n idea: A has dens. in A + 3 Slide 47 L = { x {0,1,2} n : x 21, x 20 A } {0,1,2} n A + 3 Suppose, somehow, that Slide 48 L = { x {0,1,2} n : x 21, x 20 A } {0,1,2} log log log n A + 3 Suppose, somehow, that Slide 49 L = { x {0,1,2} n : x 21, x 20 A } {0,1,2} log log log n A + 3 Suppose, somehow, that Repeat. Slide 50 L = { x {0,1,2} n : x 21, x 20 A } {0,1,2} log log log n A + 3 Suppose, somehow, that Repeat. 2 Slide 51 L = { x {0,1,2} n : x 21, x 20 A } {0,1,2} log log log n A + 3 Suppose, somehow, that Repeat. 2 Slide 52 L = { x {0,1,2} n : x 21, x 20 A } {0,1,2} log log log log log log n A + 2 3 Suppose, somehow, that Repeat. Slide 53 L = { x {0,1,2} n : x 21, x 20 A } {0,1,2} log log log log log log log log log n A + 3 3 Suppose, somehow, that Repeat. Slide 54 L = { x {0,1,2} n : x 21, x 20 A } {0,1,2} log (3/ 3 ) n A > 2/3 Suppose, somehow, that Repeat. A contains line trivially. Density Increment Argument Slide 55 L = { x {0,1,2} n : x 21, x 20 A } {0,1,2} n A + 3 Suppose, somehow, that Slide 56 L = { x {0,1,2} n : x 21, x 20 A } Suppose, somehow, that L = { x : x 21 A } { x : x 20 A } = { x : x 21 A } (harmless cheat) pretend idea: is a 12-insensitive set 12-insensitive set Slide 57 = { x : x 21 A } 12-insensitive set Slide 58 = { x : x 21 A } 12-insensitive set monkeying with 1s and 2s does not affect presence in the set Closer to an isomorphic copy of Slide 59 key thm: A dense 12-insensitive set can be almost completely partitioned into copies of Slide 60 key thm: A dense 12-insensitive set can be almost completely partitioned into copies of Slide 61 A has dens. + 3 inside A has dens. + 3 inside some Slide 62 key thm: A dense 12-insensitive set can be almost completely partitioned into copies of 2 1 1 0 1 1 1 0 1 0 2 1 1 2 1 2 2 2 1 1 1 2 lemma 1: Dense 12-insensitive set contains a copy of Slide 63 key thm: A dense 12-insensitive set can be almost completely partitioned into copies of lemma 1: Dense 12-insensitive set contains a copy of lemma 2: Dense 12-insensitive set contains a copy of {0, 1, 2} i.e., a line Slide 64 lemma 2: Dense 12-insensitive set contains a copy of {0, 1, 2} i.e., a line lemma 2: Dense 12-insensitive set contains a copy of {0, 1, 2} i.e., a line Slide 65 lemma 2: Dense 12-insensitive set contains a copy of {0, 1, 2} i.e., a line (i) Distrib. hack: make it dense in {0,1} n also. (ii) Apply DHJ(2). 1 0 1 1 1 1 1 1 0 0 0 1 0 0 0 1 1 0 1 1 A 1 0 1 1 1 1 1 1 0 1 1 1 0 0 1 1 1 1 1 1 A 1 0 1 1 1 1 1 1 0 2 2 1 0 0 2 1 1 2 1 1 A (iii) by 12-insensitivity. Slide 66 Questions?