The Hales-Jewett Theorem

Transcription

1 The Hales-Jewett Theorem A.W. HALES & R.I. JEWETT 1963 Regularity and positional games Trans. Amer. Math. Soc. 106, Hales-Jewett Theorem, SS04 p.1/35

2 Importance of HJT The Hales-Jewett theorem is presently one of the most useful techniques in Ramsey theory Hales-Jewett Theorem, SS04 p.2/35

3 Importance of HJT The Hales-Jewett theorem is presently one of the most useful techniques in Ramsey theory Without this result, Ramsey theory would more properly be called Ramseyan theorems Hales-Jewett Theorem, SS04 p.2/35

4 Outline for the lecture 1. The theorem and its consequences 1.1 Van der Waerden s Theorem 1.2 Gallai-Witt s Theorem Hales-Jewett Theorem, SS04 p.3/35

5 Outline for the lecture 1. The theorem and its consequences 1.1 Van der Waerden s Theorem 1.2 Gallai-Witt s Theorem 2. Historical Comments Hales-Jewett Theorem, SS04 p.3/35

6 Outline for the lecture 1. The theorem and its consequences 1.1 Van der Waerden s Theorem 1.2 Gallai-Witt s Theorem 2. Historical Comments 3. Proof of HJT Hales-Jewett Theorem, SS04 p.3/35

7 Basic definitions Let A be an alphabet of t symbols, usually A = {0, 1,..., t 1} Let A be a new symbol Hales-Jewett Theorem, SS04 p.4/35

8 Basic definitions Let A be an alphabet of t symbols, usually A = {0, 1,..., t 1} Let A be a new symbol Words are strings over A without Strings containing at least one are called roots Hales-Jewett Theorem, SS04 p.4/35

9 Basic definitions Let A be an alphabet of t symbols, usually A = {0, 1,..., t 1} Let A be a new symbol Words are strings over A without Strings containing at least one are called roots For a root τ (A { }) n and a symbol a A, we write τ(a) A n for the word obtained from τ by replacing each by a Hales-Jewett Theorem, SS04 p.4/35

10 Example Let A = {0, 1, 2} be an alphabet of 3 symbols. Hales-Jewett Theorem, SS04 p.5/35

11 Example Let A = {0, 1, 2} be an alphabet of 3 symbols. Some roots: τ 1 = 01 2 τ 2 = 1 τ 3 = τ i (A { }) n i, n 1 = 4, n 2 = 2, n 3 = 6 Hales-Jewett Theorem, SS04 p.5/35

12 Example Let A = {0, 1, 2} be an alphabet of 3 symbols. Some roots: τ 1 = 01 2 τ 2 = 1 τ 3 = τ i (A { }) n i, n 1 = 4, n 2 = 2, n 3 = 6 Some words obtained from these roots: τ 1 (0) = Hales-Jewett Theorem, SS04 p.5/35

13 Example Let A = {0, 1, 2} be an alphabet of 3 symbols. Some roots: τ 1 = 01 2 τ 2 = 1 τ 3 = τ i (A { }) n i, n 1 = 4, n 2 = 2, n 3 = 6 Some words obtained from these roots: τ 1 (0) = 0102 τ 1 (1) = Hales-Jewett Theorem, SS04 p.5/35

14 Example Let A = {0, 1, 2} be an alphabet of 3 symbols. Some roots: τ 1 = 01 2 τ 2 = 1 τ 3 = τ i (A { }) n i, n 1 = 4, n 2 = 2, n 3 = 6 Some words obtained from these roots: τ 1 (0) = 0102 τ 1 (1) = 0112 τ 2 (1) = Hales-Jewett Theorem, SS04 p.5/35

15 Example Let A = {0, 1, 2} be an alphabet of 3 symbols. Some roots: τ 1 = 01 2 τ 2 = 1 τ 3 = τ i (A { }) n i, n 1 = 4, n 2 = 2, n 3 = 6 Some words obtained from these roots: τ 1 (0) = 0102 τ 1 (1) = 0112 τ 2 (1) = 11 or τ 3 (2) = Hales-Jewett Theorem, SS04 p.5/35

16 Example Let A = {0, 1, 2} be an alphabet of 3 symbols. Some roots: τ 1 = 01 2 τ 2 = 1 τ 3 = τ i (A { }) n i, n 1 = 4, n 2 = 2, n 3 = 6 Some words obtained from these roots: τ 1 (0) = 0102 τ 1 (1) = 0112 τ 2 (1) = 11 or τ 3 (2) = Hales-Jewett Theorem, SS04 p.5/35

17 Further definition Let τ be a root. A combinatorial line rooted in τ is the set of t words L τ = {τ(0), τ(1),..., τ(t 1)}. Hales-Jewett Theorem, SS04 p.6/35

18 Further definition Let τ be a root. A combinatorial line rooted in τ is the set of t words L τ = {τ(0), τ(1),..., τ(t 1)}. Example: Let τ = be a root in A = {0, 1, 2, 3, 4}. Then Hales-Jewett Theorem, SS04 p.6/35

19 Further definition Let τ be a root. A combinatorial line rooted in τ is the set of t words L τ = {τ(0), τ(1),..., τ(t 1)}. Example: Let τ = be a root in A = {0, 1, 2, 3, 4}. Then L τ = L τ = 5. Hales-Jewett Theorem, SS04 p.6/35

20 The number of combinatorial lines in A n is Proof: ( A + 1) n A n Hales-Jewett Theorem, SS04 p.7/35

21 The number of combinatorial lines in A n is ( A + 1) n A n Proof: #{comb. lines in A n } = #{roots in (A { }) n } Hales-Jewett Theorem, SS04 p.7/35

22 The number of combinatorial lines in A n is ( A + 1) n A n Proof: #{comb. lines in A n } = #{roots in (A { }) n } = #{strings with at least one in (A { }) n } Hales-Jewett Theorem, SS04 p.7/35

23 The number of combinatorial lines in A n is ( A + 1) n A n Proof: #{comb. lines in A n } = #{roots in (A { }) n } = #{strings with at least one in (A { }) n } = #{all strings in (A { }) n } #{strings without } Hales-Jewett Theorem, SS04 p.7/35

24 The number of combinatorial lines in A n is ( A + 1) n A n Proof: #{comb. lines in A n } = #{roots in (A { }) n } = #{strings with at least one in (A { }) n } = #{all strings in (A { }) n } #{strings without } = ( A + 1) n A n Hales-Jewett Theorem, SS04 p.7/35

25 Why combinatorial line? Let A := {0, 1, 2}, n=3 and τ = (, 2, ). Then L τ = Hales-Jewett Theorem, SS04 p.8/35

26 Why combinatorial line? Let A := {0, 1, 2}, n=3 and τ = (, 2, ). Then L τ = x 3 x 2 x 1 Hales-Jewett Theorem, SS04 p.8/35

27 Why combinatorial line? Let A := {0, 1, 2}, n=3 and τ = (, 2, ). Then L τ = x 1 x 3 (2,2,2) (1,2,1) (0,2,0) x 2 Hales-Jewett Theorem, SS04 p.8/35

28 Theorem (HJT, 1963) Given r > 0 distinct colors and a finite alphabet A of t = A symbols. There is a dimension n = HJ(r, t) N such that for every r-coloring of the cube A n monochromatic combinatorial line. there exists a Hales-Jewett Theorem, SS04 p.9/35

29 Examples Take r = #{colors} = 1. Then for any alphabet A with t symbols HJ(1, t) = Hales-Jewett Theorem, SS04 p.10/35

30 Examples Take r = #{colors} = 1. Then for any alphabet A with t symbols HJ(1, t) = 1 Hales-Jewett Theorem, SS04 p.10/35

31 Examples Take r = #{colors} = 1. Then for any alphabet A with t symbols HJ(1, t) = 1 Let A = {0, 1} (A n then corresponds to Q n ). r = 2: HJ(2, 2) = Hales-Jewett Theorem, SS04 p.10/35

32 Examples Take r = #{colors} = 1. Then for any alphabet A with t symbols HJ(1, t) = 1 Let A = {0, 1} (A n then corresponds to Q n ). r = 2: HJ(2, 2) = 2 r = 3: HJ(3, 2) = Hales-Jewett Theorem, SS04 p.10/35

33 Examples Take r = #{colors} = 1. Then for any alphabet A with t symbols HJ(1, t) = 1 Let A = {0, 1} (A n then corresponds to Q n ). r = 2: HJ(2, 2) = 2 r = 3: HJ(3, 2) = 3 Hales-Jewett Theorem, SS04 p.10/35

34 Van der Waerden (1927) Theorem: Given r > 0 distinct colors and t N. There is a N = W (r, t) N such that for every r-coloring of the set {1,..., N} there exists at least one monochromatic arithmetic progression of t terms: { 1,..., a,..., a + d,..., a + 2d,..., a + (t 1)d,..., N } Hales-Jewett Theorem, SS04 p.11/35

35 Proof (using HJT) Let N := n(t 1) + 1 where n = HJ(r, t) from HJT. Hales-Jewett Theorem, SS04 p.12/35

36 Proof (using HJT) Let N := n(t 1) + 1 where n = HJ(r, t) from HJT. Set A := {0, 1,..., t 1} and define for x = (x 1,..., x n ) A n the mapping f : A n {1, 2,..., N} x x 1 + x x n + 1. Hales-Jewett Theorem, SS04 p.12/35

37 Proof (using HJT) Let N := n(t 1) + 1 where n = HJ(r, t) from HJT. Set A := {0, 1,..., t 1} and define for x = (x 1,..., x n ) A n the mapping f : A n {1, 2,..., N} x x 1 + x x n + 1. Thus, f induces a coloring of A n color of x A n := color of number f(x). Hales-Jewett Theorem, SS04 p.12/35

38 Proof (cont.) Every combinatorial line L τ = {τ(0), τ(1),..., τ(t 1)} is mapped to an arithmetic progression of length t. Hales-Jewett Theorem, SS04 p.13/35

39 Proof (cont.) Every combinatorial line L τ = {τ(0), τ(1),..., τ(t 1)} is mapped to an arithmetic progression of length t. a := f(τ(0)) + 1 d := #{ s in τ} Hales-Jewett Theorem, SS04 p.13/35

40 Proof (cont.) Every combinatorial line L τ = {τ(0), τ(1),..., τ(t 1)} is mapped to an arithmetic progression of length t. a := f(τ(0)) + 1 d := #{ s in τ} By HJT there is a monochromatic line in A n, which corresponds to a monochromatic arithmetic progression of length t. Hales-Jewett Theorem, SS04 p.13/35

41 Gallai-Witt (1943,1951) A set of vectors U Z m is a homothetic copy of V Z m if there is a vector u Z m and a constant λ N, such that U = u + λv := {u + λv v V } Hales-Jewett Theorem, SS04 p.14/35

42 Gallai-Witt (1943,1951) A set of vectors U Z m is a homothetic copy of V Z m if there is a vector u Z m and a constant λ N, such that U = u + λv := {u + λv v V } Theorem: Given r > 0 distinct colors and let the vectors of Z m be r-colored. Then every finite subset V copy which is monochromatic. Z m has a homothetic Hales-Jewett Theorem, SS04 p.14/35

43 Proof (using HJT) Let r := #{colors} and V := {v 0,..., v t 1 } = A. Set n = HJ(r, t) and consider A n x = (x 1,..., x n ) x i {v 0,..., v t 1 }. Hales-Jewett Theorem, SS04 p.15/35

44 Proof (using HJT) Let r := #{colors} and V := {v 0,..., v t 1 } = A. Set n = HJ(r, t) and consider A n x = (x 1,..., x n ) x i {v 0,..., v t 1 }. Define f : A n Z m by f(x) = x x n. This induces a r-coloring of A n. Hales-Jewett Theorem, SS04 p.15/35

45 Proof (using HJT) Let r := #{colors} and V := {v 0,..., v t 1 } = A. Set n = HJ(r, t) and consider A n x = (x 1,..., x n ) x i {v 0,..., v t 1 }. Define f : A n Z m by f(x) = x x n. This induces a r-coloring of A n. By HJT there is a monochromatic combinatorial line L τ = {τ(v 0 ),..., τ(v t 1 )} A n. Hales-Jewett Theorem, SS04 p.15/35

46 Proof (cont.) Let I = {i τ i }, and λ := #{ s in τ} = n I > 0. Hales-Jewett Theorem, SS04 p.16/35

47 Proof (cont.) Let I = {i τ i }, and λ := #{ s in τ} = n I > 0. Then f(l τ ) = { u + λv j u = i I } v i ; j = 0,..., t 1 Hales-Jewett Theorem, SS04 p.16/35

48 Proof (cont.) Let I = {i τ i }, and λ := #{ s in τ} = n I > 0. Then f(l τ ) = { u + λv j u = i I } v i ; j = 0,..., t 1 This is a homothetic copy of V = {v 0,..., v t 1 }. Hales-Jewett Theorem, SS04 p.16/35

49 Historical comments Van der Waerden s Theorem was originally conjectured by Baudet. The theorem is a corollary of Szemerédi s theorem. Hales-Jewett Theorem, SS04 p.17/35

50 Historical comments Van der Waerden s Theorem was originally conjectured by Baudet. The theorem is a corollary of Szemerédi s theorem. Szemerédi s theorem was conjectured in 1936 by Erdos and Turán. Roth (1953) proved the case t = 3 (mentioned in his Fields Medal citation). Szemerédi proved the case t = 4 (1969), and the general theorem in Hales-Jewett Theorem, SS04 p.17/35

51 Historical comments Van der Waerden s Theorem was originally conjectured by Baudet. The theorem is a corollary of Szemerédi s theorem. Szemerédi s theorem was conjectured in 1936 by Erdos and Turán. Roth (1953) proved the case t = 3 (mentioned in his Fields Medal citation). Szemerédi proved the case t = 4 (1969), and the general theorem in Fürstenberg and Katznelson proved Szemerédi s theorem using ergodic theory in Hales-Jewett Theorem, SS04 p.17/35

52 Historical comments Van der Waerden s Theorem was originally conjectured by Baudet. The theorem is a corollary of Szemerédi s theorem. Szemerédi s theorem was conjectured in 1936 by Erdos and Turán. Roth (1953) proved the case t = 3 (mentioned in his Fields Medal citation). Szemerédi proved the case t = 4 (1969), and the general theorem in Fürstenberg and Katznelson proved Szemerédi s theorem using ergodic theory in Gowers gave a new proof (1998), with a better bound on W (r, t), for the case t = 4 (mentioned in his Fields Medal citation). Hales-Jewett Theorem, SS04 p.17/35

53 Known W (r, t) The following table shows all exactly known van der Waerden numbers (the nontrivials marked red) r/t t = 1 t = 2 t = 3 t = 4 t = 5 r = r = r = r = Eric W. Weisstein et al. van der Waerden Number. From MathWorld A Wolfram Web Resource. Hales-Jewett Theorem, SS04 p.18/35

54 Some bounds It is known that and for some constants c 1, c 2. W (r, 3) e rc 1 c 2 W (r, 4) e eer Eric W. Weisstein et al. van der Waerden Number. From MathWorld A Wolfram Web Resource. Hales-Jewett Theorem, SS04 p.19/35

55 History of HJT Various proofs of the Hales-Jewett theorem are known. The original proof of Hales and Jewett is quite short but provides an extremely fast growing upper bound for HJ(r, t). Hales-Jewett Theorem, SS04 p.20/35

56 History of HJT Various proofs of the Hales-Jewett theorem are known. The original proof of Hales and Jewett is quite short but provides an extremely fast growing upper bound for HJ(r, t). Shelah (1988) found a completely new proof which yields a much smaller (a primitive recursive) bound. A function that can be implemented using only for-loops (which have a fixed iteration limit) is called primitive recursive. Hales-Jewett Theorem, SS04 p.20/35

57 History of HJT Various proofs of the Hales-Jewett theorem are known. The original proof of Hales and Jewett is quite short but provides an extremely fast growing upper bound for HJ(r, t). Shelah (1988) found a completely new proof which yields a much smaller (a primitive recursive) bound. A function that can be implemented using only for-loops (which have a fixed iteration limit) is called primitive recursive. We will follow the compact version of Shelah s proof from A. Nilli (1990) (c/o Noga Alon). Hales-Jewett Theorem, SS04 p.20/35

58 Proof (of HJT) By induction on t = A. Let r := #{colors}. Hales-Jewett Theorem, SS04 p.21/35

59 Proof (of HJT) By induction on t = A. Let r := #{colors}. t = 1: trivial, HJ(r, 1) = 1 t 1 t: Assume HJT holds for t 1 (and r). Hales-Jewett Theorem, SS04 p.21/35

60 Proof (of HJT) By induction on t = A. Let r := #{colors}. t = 1: trivial, HJ(r, 1) = 1 t 1 t: Assume HJT holds for t 1 (and r). Set n := HJ(r, t 1) and define a sequence N 1 <... < N n N 1 := r tn N i := r tn + i 1 j=1 N j i 2 Hales-Jewett Theorem, SS04 p.21/35

61 Proof (of HJT) By induction on t = A. Let r := #{colors}. t = 1: trivial, HJ(r, 1) = 1 t 1 t: Assume HJT holds for t 1 (and r). Set n := HJ(r, t 1) and define a sequence N 1 <... < N n N 1 := r tn N i := r tn + i 1 j=1 N j i 2 Define as a new Dimension N := N N n. Hales-Jewett Theorem, SS04 p.21/35

62 We want to prove that for every coloring χ of the N-cube A N χ : A N {1,..., r} there is at least one monochromatic combinatorial line, that is, HJ(r, t) N. Hales-Jewett Theorem, SS04 p.22/35

63 We want to prove that for every coloring χ of the N-cube A N χ : A N {1,..., r} there is at least one monochromatic combinatorial line, that is, HJ(r, t) N. How large is this N? Hales-Jewett Theorem, SS04 p.22/35

64 We want to prove that for every coloring χ of the N-cube A N χ : A N {1,..., r} there is at least one monochromatic combinatorial line, that is, HJ(r, t) N. How large is this N? We have n = HJ(r, t 1). Estimate N N > N n > t n. r r r.. }{{} n times Hales-Jewett Theorem, SS04 p.22/35

65 We want to prove that for every coloring χ of the N-cube A N χ : A N {1,..., r} there is at least one monochromatic combinatorial line, that is, HJ(r, t) N. How large is this N? We have n = HJ(r, t 1). Estimate N N > N n > t n. r r r.. }{{} n times Example: Take r = 2 and t = 3. n = HJ(2, 2) = 2. Then N > = Hales-Jewett Theorem, SS04 p.22/35

66 Definition: Neighbors a, b A n are neighbors if they differ in exactly one coordinate. Hales-Jewett Theorem, SS04 p.23/35

67 Definition: Neighbors a, b A n are neighbors if they differ in exactly one coordinate. Example: a, b A n differ in the i-th coordinate a = a 1... a i 1 0 a i+1... a n b = a 1... a i 1 1 a i+1... a n Hales-Jewett Theorem, SS04 p.23/35

68 Definition: Neighbors a, b A n are neighbors if they differ in exactly one coordinate. Example: a, b A n differ in the i-th coordinate a = a 1... a i 1 0 a i+1... a n b = a 1... a i 1 1 a i+1... a n For a = a 1... a n A n and a concatenation of n roots denote τ = τ 1... τ n τ i = N i i = 1,..., n τ(a) = τ 1 (a 1 )... τ n (a n ) τ(a) = N. Hales-Jewett Theorem, SS04 p.23/35

69 Claim Claim: There are n roots τ 1,..., τ n of length τ i = N i such that for their concatenation τ χ(τ(a)) = χ(τ(b)) for any two neighbors a, b A n. Hales-Jewett Theorem, SS04 p.24/35

70 Claim Claim: There are n roots τ 1,..., τ n of length τ i = N i such that for their concatenation τ χ(τ(a)) = χ(τ(b)) for any two neighbors a, b A n. How does this imply the theorem? Hales-Jewett Theorem, SS04 p.24/35

71 Claim Claim: There are n roots τ 1,..., τ n of length τ i = N i such that for their concatenation τ χ(τ(a)) = χ(τ(b)) for any two neighbors a, b A n. How does this imply the theorem? Take such a τ from the claim and define a new coloring χ for the n-cube (A {0}) n χ (a) := χ(τ(a)) χ : A N {1,..., r}. Hales-Jewett Theorem, SS04 p.24/35

72 Since A {0} = t 1 and n = HJ(r, t 1) by induction assumption there is a root ν = ν 1... ν n ( (A {0}) { } ) n such that the combinatorial line L ν = {ν(1), ν(2),..., ν(t 1)} is monochromatic with respect to χ. Hales-Jewett Theorem, SS04 p.25/35

73 Since A {0} = t 1 and n = HJ(r, t 1) by induction assumption there is a root ν = ν 1... ν n ( (A {0}) { } ) n such that the combinatorial line L ν = {ν(1), ν(2),..., ν(t 1)} is monochromatic with respect to χ. Consider the root τ(ν) = τ 1 (ν 1 )... τ n (ν n ) τ(ν) = N. Hales-Jewett Theorem, SS04 p.25/35

74 Since A {0} = t 1 and n = HJ(r, t 1) by induction assumption there is a root ν = ν 1... ν n ( (A {0}) { } ) n such that the combinatorial line L ν = {ν(1), ν(2),..., ν(t 1)} is monochromatic with respect to χ. Consider the root τ(ν) = τ 1 (ν 1 )... τ n (ν n ) τ(ν) = N. We will show that the line L τ(ν) = { τ(ν(0)), τ(ν(1)),..., τ(ν(t 1)) } is monochromatic with respect to χ. Hales-Jewett Theorem, SS04 p.25/35

75 Since L ν is monochromatic w.r.t. χ χ (ν(1)) =... = χ (ν(t 1)). Hales-Jewett Theorem, SS04 p.26/35

76 Since L ν is monochromatic w.r.t. χ χ (ν(1)) =... = χ (ν(t 1)). By definition of χ χ ( τ(ν(1)) ) =... = χ ( τ(ν(t 1)) ). What about χ ( τ(ν(0)) ) =? Hales-Jewett Theorem, SS04 p.26/35

77 Since L ν is monochromatic w.r.t. χ By definition of χ χ (ν(1)) =... = χ (ν(t 1)). χ ( τ(ν(1)) ) =... = χ ( τ(ν(t 1)) ). What about χ ( τ(ν(0)) ) =? If #{ s in ν} = 1, then τ(ν(0)) is neighbor of τ(ν(1)). By the claim χ ( τ(ν(0)) ) = χ ( τ(ν(1)) ). Hales-Jewett Theorem, SS04 p.26/35

78 If #{ s in ν} > 1, then we still can reach τ(ν(0)) from τ(ν(1)) by passing through a sequence of neighbors Hales-Jewett Theorem, SS04 p.27/35

79 If #{ s in ν} > 1, then we still can reach τ(ν(0)) from τ(ν(1)) by passing through a sequence of neighbors τ(ν(1)) = τ(ν(0)) = Hales-Jewett Theorem, SS04 p.27/35

80 If #{ s in ν} > 1, then we still can reach τ(ν(0)) from τ(ν(1)) by passing through a sequence of neighbors τ(ν(1)) = τ(ν(0)) = Again by the claim above χ ( τ(ν(0)) ) = χ ( τ(ν(1)) ). Thus, L τ(ν) is monochromatic. Hales-Jewett Theorem, SS04 p.27/35

81 Proof (of claim) We prove the existence of roots τ i by backward induction on i. Suppose we already have defined τ i+1,..., τ n. Hales-Jewett Theorem, SS04 p.28/35

82 Proof (of claim) We prove the existence of roots τ i by backward induction on i. Suppose we already have defined τ i+1,..., τ n. Recall that we want τ i = N i. Let M i 1 := i 1 j=1 N j. For k = 0,..., N i let W k be the word W k := }{{} k }{{} N i k Hales-Jewett Theorem, SS04 p.28/35

83 Proof (of claim) We prove the existence of roots τ i by backward induction on i. Suppose we already have defined τ i+1,..., τ n. Recall that we want τ i = N i. Let M i 1 := i 1 j=1 N j. For k = 0,..., N i let W k be the word W k := }{{} k }{{} N i k For each k define the r-coloring χ k of all words in A M i 1+n i as χ k ( x1... x Mi 1 y i+1... y n ) := := χ ( x 1... x Mi 1 W k τ i+1 (y i+1 )... τ n (y n ) ). Hales-Jewett Theorem, SS04 p.28/35

84 Proof (cont.) We have N i + 1 colorings χ 0,..., χ Ni. The total number of such r-colorings is r #{of words} = r tm i 1 +n i r tm i 1 +n = N i. Hales-Jewett Theorem, SS04 p.29/35

85 Proof (cont.) We have N i + 1 colorings χ 0,..., χ Ni. The total number of such r-colorings is r #{of words} = r tm i 1 +n i r tm i 1 +n = N i. By the pigeonhole principle there is χ s = χ k for some s < k. Let τ i := }{{} s... }{{} k s }{{} N i k Hales-Jewett Theorem, SS04 p.29/35

86 Proof (cont.) We have N i + 1 colorings χ 0,..., χ Ni. The total number of such r-colorings is r #{of words} = r tm i 1 +n i r tm i 1 +n = N i. By the pigeonhole principle there is χ s = χ k for some s < k. Let τ i := }{{} s... }{{} k s }{{} N i k We have to check that τ = τ 1... τ n satisfies the assertion of the claim. Hales-Jewett Theorem, SS04 p.29/35

87 Proof (cont.) Observe that τ i (0) = W k and τ i (1) = W s. Hales-Jewett Theorem, SS04 p.30/35

88 Proof (cont.) Observe that τ i (0) = W k and τ i (1) = W s. Take a, b A n neighbors differing in the i-th coordinate a = a 1... a i 1 0 a i+1... a n b = a 1... a i 1 1 a i+1... a n Hales-Jewett Theorem, SS04 p.30/35

89 Proof (cont.) Observe that τ i (0) = W k and τ i (1) = W s. Take a, b A n neighbors differing in the i-th coordinate a = a 1... a i 1 0 a i+1... a n b = a 1... a i 1 1 a i+1... a n then τ(a) = τ 1 (a 1 )... τ i 1 (a i 1 ) τ i (0) τ i+1 (a i+1 )... τ n (a n ) τ(b) = τ 1 (a 1 )... τ i 1 (a i 1 ) τ i (1) τ i+1 (a i+1 )... τ n (a n ) Hales-Jewett Theorem, SS04 p.30/35

90 Proof (cont.) Observe that τ i (0) = W k and τ i (1) = W s. Take a, b A n neighbors differing in the i-th coordinate a = a 1... a i 1 0 a i+1... a n b = a 1... a i 1 1 a i+1... a n then τ(a) = τ 1 (a 1 )... τ i 1 (a i 1 ) τ i (0) τ i+1 (a i+1 )... τ n (a n ) τ(b) = τ 1 (a 1 )... τ i 1 (a i 1 ) τ i (1) τ i+1 (a i+1 )... τ n (a n ) and since χ s = χ k Hales-Jewett Theorem, SS04 p.30/35

91 Proof (cont.) χ(τ(a)) = χ ( τ 1 (a 1 )... τ i 1 (a i 1 ) τ i (0) τ i+1 (a i+1 )... τ n (a n ) ) Hales-Jewett Theorem, SS04 p.31/35

92 Proof (cont.) χ(τ(a)) = χ ( τ 1 (a 1 )... τ i 1 (a i 1 ) τ i (0) τ i+1 (a i+1 )... τ n (a n ) ) = χ ( τ 1 (a 1 )... τ i 1 (a i 1 ) W k τ i+1 (a i+1 )... τ n (a n ) ) Hales-Jewett Theorem, SS04 p.31/35

93 Proof (cont.) χ(τ(a)) = χ ( τ 1 (a 1 )... τ i 1 (a i 1 ) τ i (0) τ i+1 (a i+1 )... τ n (a n ) ) = χ ( τ 1 (a 1 )... τ i 1 (a i 1 ) W k τ i+1 (a i+1 )... τ n (a n ) ) = χ k ( τ1 (a 1 )... τ i 1 (a i 1 ) a i+1... a n ) Hales-Jewett Theorem, SS04 p.31/35

94 Proof (cont.) χ(τ(a)) = χ ( τ 1 (a 1 )... τ i 1 (a i 1 ) τ i (0) τ i+1 (a i+1 )... τ n (a n ) ) = χ ( τ 1 (a 1 )... τ i 1 (a i 1 ) W k τ i+1 (a i+1 )... τ n (a n ) ) ( ) = χ k τ1 (a 1 )... τ i 1 (a i 1 ) a i+1... a n ( ) = χ s τ1 (a 1 )... τ i 1 (a i 1 ) a i+1... a n Hales-Jewett Theorem, SS04 p.31/35

95 Proof (cont.) χ(τ(a)) = χ ( τ 1 (a 1 )... τ i 1 (a i 1 ) τ i (0) τ i+1 (a i+1 )... τ n (a n ) ) = χ ( τ 1 (a 1 )... τ i 1 (a i 1 ) W k τ i+1 (a i+1 )... τ n (a n ) ) ( ) = χ k τ1 (a 1 )... τ i 1 (a i 1 ) a i+1... a n ( ) = χ s τ1 (a 1 )... τ i 1 (a i 1 ) a i+1... a n = χ ( τ 1 (a 1 )... τ i 1 (a i 1 ) W s τ i+1 (a i+1 )... τ n (a n ) ) Hales-Jewett Theorem, SS04 p.31/35

96 Proof (cont.) χ(τ(a)) = χ ( τ 1 (a 1 )... τ i 1 (a i 1 ) τ i (0) τ i+1 (a i+1 )... τ n (a n ) ) = χ ( τ 1 (a 1 )... τ i 1 (a i 1 ) W k τ i+1 (a i+1 )... τ n (a n ) ) ( ) = χ k τ1 (a 1 )... τ i 1 (a i 1 ) a i+1... a n ( ) = χ s τ1 (a 1 )... τ i 1 (a i 1 ) a i+1... a n = χ ( τ 1 (a 1 )... τ i 1 (a i 1 ) W s τ i+1 (a i+1 )... τ n (a n ) ) = χ ( τ 1 (a 1 )... τ i 1 (a i 1 ) τ i (1) τ i+1 (a i+1 )... τ n (a n ) ) Hales-Jewett Theorem, SS04 p.31/35

97 Proof (cont.) χ(τ(a)) = χ ( τ 1 (a 1 )... τ i 1 (a i 1 ) τ i (0) τ i+1 (a i+1 )... τ n (a n ) ) = χ ( τ 1 (a 1 )... τ i 1 (a i 1 ) W k τ i+1 (a i+1 )... τ n (a n ) ) ( ) = χ k τ1 (a 1 )... τ i 1 (a i 1 ) a i+1... a n ( ) = χ s τ1 (a 1 )... τ i 1 (a i 1 ) a i+1... a n = χ ( τ 1 (a 1 )... τ i 1 (a i 1 ) W s τ i+1 (a i+1 )... τ n (a n ) ) = χ ( τ 1 (a 1 )... τ i 1 (a i 1 ) τ i (1) τ i+1 (a i+1 )... τ n (a n ) ) = χ(τ(b)) This completes the proof of the claim. Hales-Jewett Theorem, SS04 p.31/35

98 Combinatorial m-space Let τ be a root in (A { 1,..., m }) n, where 1,..., m A are distinct symbols. We require that each of these appears at least once in τ. Thus, we have m mutually disjoint sets of moving coordinates. Hales-Jewett Theorem, SS04 p.32/35

99 Combinatorial m-space Let τ be a root in (A { 1,..., m }) n, where 1,..., m A are distinct symbols. We require that each of these appears at least once in τ. Thus, we have m mutually disjoint sets of moving coordinates. The combinatorial m-space S τ is the set of all t m words in A n obtained by replacing each i with a symbol from A. Hales-Jewett Theorem, SS04 p.32/35

100 Combinatorial m-space Let τ be a root in (A { 1,..., m }) n, where 1,..., m A are distinct symbols. We require that each of these appears at least once in τ. Thus, we have m mutually disjoint sets of moving coordinates. The combinatorial m-space S τ is the set of all t m words in A n obtained by replacing each i with a symbol from A. Remark: A combinatorial line is a combinatorial 1- space. Hales-Jewett Theorem, SS04 p.32/35

101 Example A combinatorial 2-space in A 3 = {0, 1, 2} 3. x 3 x 1 Here τ = ( 1, 1, 2 ). x 2 Hales-Jewett Theorem, SS04 p.33/35

102 Theorem (generalized) Given r > 0 distinct colors, a dimension m N and a finite alphabet A of t = A symbols. There is a dimension n = HJ(m, r, t) N such that for every r-coloring of the cube A n there exists a monochromatic combinatorial m-space. Hales-Jewett Theorem, SS04 p.34/35

103 End Hales-Jewett Theorem, SS04 p.35/35