Presentation is loading. Please wait.

Presentation is loading. Please wait.

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

Similar presentations


Presentation on theme: "Presented by: Ryan ODonnell Carnegie Melloni by D. H. J. Polymath."— Presentation transcript:

1 presented by: Ryan ODonnell Carnegie Melloni by D. H. J. Polymath

2

3

4 If A {0,1,2} n has density Ω(1), then A contains a (combinatorial) line. DHJ(3): point: line: { },,

5 [Furstenberg-Katznelson91] : DHJ(k) is true k. (k = 3): δ > 0, n 0 (δ) s.t. n n 0 (δ), A {0,1,2} n has density δ A contains a line proof: Used ergodic theory. No effective bound on n 0 (δ).

6 Importance in combinatorics: Implies Szemerédis Theorem

7 Why I was interested: , 1, 2, A {0,1,2} n, no trips w/ all cols , 0, 1, 0, A {0,1} n, no trips w/ all cols see [O-Wu09]

8

9

10

11

12

13

14

15

16

17

18

19

20

21 [D.H.J. Polymath]: δ > 0, n 2 O(1/δ 3 ), A {0,1,2} n has density δ A contains a line proof: Elementary probability/combinatorics. also the general k case, with Ackermann-type bounds

22 Proof sketch

23 DHJ(2): A {0,1} n dens. δ A contains line line: { }, contrapositive: If A has no comparable pairs i.e., A is an antichain then A is very sparse. Sperners Theorem

24 DHJ(2): A {0,1} n dens. δ A contains line n equal-slices distribution

25 DHJ(2): A {0,1} n dens. δ A contains line equal-slices distribution 5 0 n n (each Hamming wt. equally likely)

26 DHJ(2): A {0,1} n dens. δ A contains line 5 0 n n equal-slices distribution DHJ(2): A {0,1} n eq-slices dens. δ A contains line

27 5 0 n n DHJ(2): A {0,1} n eq-slices dens. δ A contains line

28 5 0 n n DHJ(2): A {0,1} n eq-slices dens. δ A contains line

29 5 0 n n DHJ(2): A {0,1} n eq-slices dens. δ A contains line

30 5 0 n DHJ(2): A {0,1} n eq-slices dens. δ A contains line ( Pr[degen] = ) random eq-slices line

31 DHJ(2): A {0,1} n eq-slices dens. δ eq-slice-line in A w.p. δ 2 Pr[ x, y A ] = E xy x y A A & Pr = E x x A 2 Pr E x x A 2 = δ 2 (independent)

32 DHJ(2): A {0,1} n eq-slices dens. δ eq-slice-line in A w.p. δ 2

33 DHJ(2): A {0,1} n eq-slices dens. δ n 1/δ 2 A contains a line

34 DHJ(2): A {0,1} n eq-slices dens. δ Distribution Hackery unif rand. coords unif on eq-slices on

35 [ eq-slices ( A ) ] DHJ(2): A {0,1} n eq-slices dens. δ Distribution Hackery unif easy: d TV ( unif, unif ) = o(1)unif (A) δ = E [ eq-slices A y y rand. coords

36 5 0 n n DHJ(2): A {0,1} n eq-slices dens. δ eq-slice-line in A w.p. δ 2

37 DHJ(3): A {0,1,2} n eq-slices dens. δ eq-slice-line in A w.p. δ 3 ???

38 Pr[ x, y A ] = E xy x y A A & Pr = E x y x A 2 Pr E xy x A 2 = δ 2 (independent)

39 5 0 n DHJ(3): A {0,1,2} n eq-slices dens. δ ? n

40 5 0 n n DHJ(3): A {0,1,2} n eq-slices dens. δ 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!

41 DHJ(3): A {0,1,2} n eq-slices dens. δ by distrib. hackery: A {0,1} n eq-slices dens. δ also {0,1,2} n {0,1} n A A

42 DHJ(3): A {0,1,2} n eq-slices dens. δ by distrib. hackery: A {0,1} n eq-slices dens. δ also {0,1,2} n {0,1} n A A L L = { x {0,1,2} n : x 21, x 20 A } δ 2

43 L = { x {0,1,2} n : x 21, x 20 A } {0,1,2} n {0,1} n A A L δ 2 L = { x {0,1,2} n : x 21, x 20 A } {0,1,2} n {0,1} n A A L δ 2

44 L = { x {0,1,2} n : x 21, x 20 A } {0,1,2} n {0,1} n A A L δ 2 if L A, done! δ

45 L = { x {0,1,2} n : x 21, x 20 A } {0,1,2} n {0,1} n A A L δ 2 δ idea: A has dens. in

46 L = { x {0,1,2} n : x 21, x 20 A } {0,1,2} n idea: A has dens. in A δ + δ 3

47 L = { x {0,1,2} n : x 21, x 20 A } {0,1,2} n A δ + δ 3 Suppose, somehow, that

48 L = { x {0,1,2} n : x 21, x 20 A } {0,1,2} log log log n A δ + δ 3 Suppose, somehow, that

49 L = { x {0,1,2} n : x 21, x 20 A } {0,1,2} log log log n A δ + δ 3 Suppose, somehow, that Repeat.

50 L = { x {0,1,2} n : x 21, x 20 A } {0,1,2} log log log n A δ + δ 3 Suppose, somehow, that Repeat. δ 2

51 L = { x {0,1,2} n : x 21, x 20 A } {0,1,2} log log log n A δ + δ 3 Suppose, somehow, that Repeat. δ 2

52 L = { x {0,1,2} n : x 21, x 20 A } {0,1,2} log log log log log log n A δ + 2δ 3 Suppose, somehow, that Repeat.

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 δ + 3δ 3 Suppose, somehow, that Repeat.

54 L = { x {0,1,2} n : x 21, x 20 A } {0,1,2} log (3/δ 3 ) n A > 2/3 Suppose, somehow, that Repeat. A contains line trivially. Density Increment Argument

55 L = { x {0,1,2} n : x 21, x 20 A } {0,1,2} n A δ + δ 3 Suppose, somehow, that

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

57 = { x : x 21 A } 12-insensitive set

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

59 key thm: A dense 12-insensitive set can be almost completely partitioned into copies of

60 key thm: A dense 12-insensitive set can be almost completely partitioned into copies of

61 A has dens. δ + δ 3 inside A has dens. δ + δ 3 inside some

62 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

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

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

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) A A A (iii) … by 12-insensitivity.

66 Questions?


Download ppt "Presented by: Ryan ODonnell Carnegie Melloni by D. H. J. Polymath."

Similar presentations


Ads by Google