A disjoint union theorem for trees

Transcription

1 University of Warwick Mathematics Institute Fields Institute, 05

2 Finite disjoint union Theorem Theorem (Folkman) For every pair of positive integers m and r there is integer n 0 such that for every r-coloring of the power-set P(X) of some set X of cardinality at least n 0, there is a family D = (D i ) m i= of pairwise disjoint nonempty subsets of X such that the family { } U(D) = D i : I {,,..., m} i I of non-empty unions is monochromatic.

3 Infinite disjoint union Theorem Theorem (Carlson-Simpson) For every finite Souslin measurable coloring of the power-set P(ω) of ω, there is a sequence D = (D n ) n<ω of pairwise disjoint subsets of the natural numbers such that the set { } U(D) = D n : M is a non-empty subset of ω is monochromatic. n M

4 Trees A tree is a partially ordered set (T, T ) such that Pred T (t) = {s T : s < T t} is is finite and totally ordered for all t in T. We consider only uniquely rooted and finitely branching trees with no maximal nodes.

5 Levels For n < ω, the n-th level of T, is the set T(n) = {t T : Pred T (t) = n}. T (3) T () T () T (0)

6 Level set For a subset D of T, we define its level set L T (D) = {n ω : D T(n) } L T (D) = {, 3}

7 Vector trees From now on, fix an integer d. A vector tree T = (T,..., T d ) is a d-sequence of uniquely rooted and finitely branching trees with no maximal nodes. T T T d

8 Level products For a vector tree T = (T,..., T d ) we define its level product as T = T (n)... T d (n) n<ω The n-th level of the level product of T is T(n) = T (n)... T d (n). T(3) T() T() T(0) T T T d

9 Vector trees Let T = (T,..., T d ) a vector tree. For t = (t,..., t d ) and s = (s,..., s d ) in T, set t T s iff t i Ti s i for all i =,..., d. For t = (t,..., t d ) in T, we define Succ T (t) = {s T : t T s}

10 Vector subsets and dense vector subsets products A sequence D = (D,..., D d ) is called a vector subset of T if D i is a subset of T i for all i =,..., d and L T (D ) =... = L Td (D d ). For a vector subset D of T we define its level product D = (T (n) D )... (T d (n) D d ). n<ω For t T, a vector subset D of T is t-dense,, ( n)( m)( s T(n) Succ T (t)( s T(m) D) s T s. D is called dense if it is root( T)-dense.

11 Vector subsets and dense vector subsets products A sequence D = (D,..., D d ) is called a vector subset of T if D i is a subset of T i for all i =,..., d and L T (D ) =... = L Td (D d ). For a vector subset D of T we define its level product D = (T (n) D )... (T d (n) D d ). n<ω For t T, a vector subset D of T is t-dense,, ( n)( m)( s T(n) Succ T (t)( s T(m) D) s T s. D is called dense if it is root( T)-dense.

12 Dense vector subset D D D t ( n)( m)( s T(n) Succ T (t)( s T(m) D) s T s. T

13 The Halpern Läuchli Theorem Theorem (Halpern Läuchli) Let T be a vector tree. Then for every dense vector subset D of T and every subset P of D, there exists a vector subset D of D such that either (i) D is a subset of P and D is a dense vector subset of T, or (ii) D is a subset of P c and D is a t-dense vector subset of T for some t in T.

14 Subspaces Let T be a vector tree. We define U(T) = {U T : U has a minimum}. We let U(T) take its topology from {0, } T. Let D be a vector subset of T. A D-subspace of U(T) is a family U = (U t ) t D such that U t U(T) for all t D, U s U t = for s t, 3 min U t = t for all t D.

15 The span of a subspace For a subspace U = (U t ) t D(U) we define its span by { [U] = t Γ { = t Γ } U t : Γ D(U) U(T) } U t : Γ D(U) and Γ U(T). If U and U are two subspaces of U(T), we say that U is a subspace of U, and write U U, if [U ] [U]. Remark U U implies that D(U ) is a vector subset of D(U).

16 Disjoint union Theorem for vector trees Theorem Let T be a vector tree and P a Souslin measurable subset of U(T). Also let D be a dense vector subset of T and U a D-subspace of U(T). Then there exists a subspace U of U(T) with U U such that either (i) [U ] is a subset of P and D(U ) is a dense vector subset of T, or (ii) [U ] is a subset of P c and D(U ) is a t-dense vector subset of T for some t in T.

17 Corollary (Carlson Simpson) For every finite Souslin measurable coloring of P(ω) there is a sequence D = (D n ) n<ω of pairwise disjoint subsets of ω such that the set U(D) is monochromatic. Let Λ be a finite alphabet. We view the elements of Λ ω as infinite constant words over Λ. Also let (v n ) n be a sequence of distinct symbols that do not occur in Λ. An infinite dimensional variable word is a map f : ω Λ {v n : n N} such that for every n we have that f (v n ) and max f (v n ) < min f (v n+ ). If (a n ) n Λ ω then by f ((a n ) n ) we denote the constant word resulting by substituting each occurrence of v n by a n. Theorem Let Λ be a finite alphabet. Then for every Souslin measurable coloring of Λ ω there exists an infinite dimensional word such that the set {f ((a n ) n ) : (a n ) n Λ ω } is monochromatic.

18 Corollary (Carlson Simpson) For every finite Souslin measurable coloring of P(ω) there is a sequence D = (D n ) n<ω of pairwise disjoint subsets of ω such that the set U(D) is monochromatic. Let Λ be a finite alphabet. We view the elements of Λ ω as infinite constant words over Λ. Also let (v n ) n be a sequence of distinct symbols that do not occur in Λ. An infinite dimensional variable word is a map f : ω Λ {v n : n N} such that for every n we have that f (v n ) and max f (v n ) < min f (v n+ ). If (a n ) n Λ ω then by f ((a n ) n ) we denote the constant word resulting by substituting each occurrence of v n by a n. Theorem Let Λ be a finite alphabet. Then for every Souslin measurable coloring of Λ ω there exists an infinite dimensional word such that the set {f ((a n ) n ) : (a n ) n Λ ω } is monochromatic.

19 Corollary (Carlson Simpson) For every finite Souslin measurable coloring of P(ω) there is a sequence D = (D n ) n<ω of pairwise disjoint subsets of ω such that the set U(D) is monochromatic. Let Λ be a finite alphabet. We view the elements of Λ ω as infinite constant words over Λ. Also let (v n ) n be a sequence of distinct symbols that do not occur in Λ. An infinite dimensional variable word is a map f : ω Λ {v n : n N} such that for every n we have that f (v n ) and max f (v n ) < min f (v n+ ). If (a n ) n Λ ω then by f ((a n ) n ) we denote the constant word resulting by substituting each occurrence of v n by a n. Theorem Let Λ be a finite alphabet. Then for every Souslin measurable coloring of Λ ω there exists an infinite dimensional word such that the set {f ((a n ) n ) : (a n ) n Λ ω } is monochromatic.

20 Corollary (Carlson Simpson) For every finite Souslin measurable coloring of P(ω) there is a sequence D = (D n ) n<ω of pairwise disjoint subsets of ω such that the set U(D) is monochromatic. Let Λ be a finite alphabet. We view the elements of Λ ω as infinite constant words over Λ. Also let (v n ) n be a sequence of distinct symbols that do not occur in Λ. An infinite dimensional variable word is a map f : ω Λ {v n : n N} such that for every n we have that f (v n ) and max f (v n ) < min f (v n+ ). If (a n ) n Λ ω then by f ((a n ) n ) we denote the constant word resulting by substituting each occurrence of v n by a n. Theorem Let Λ be a finite alphabet. Then for every Souslin measurable coloring of Λ ω there exists an infinite dimensional word such that the set {f ((a n ) n ) : (a n ) n Λ ω } is monochromatic.

21 Corollary (Carlson Simpson) For every finite Souslin measurable coloring of P(ω) there is a sequence D = (D n ) n<ω of pairwise disjoint subsets of ω such that the set U(D) is monochromatic. Let Λ be a finite alphabet. We view the elements of Λ ω as infinite constant words over Λ. Also let (v n ) n be a sequence of distinct symbols that do not occur in Λ. An infinite dimensional variable word is a map f : ω Λ {v n : n N} such that for every n we have that f (v n ) and max f (v n ) < min f (v n+ ). If (a n ) n Λ ω then by f ((a n ) n ) we denote the constant word resulting by substituting each occurrence of v n by a n. Theorem Let Λ be a finite alphabet. Then for every Souslin measurable coloring of Λ ω there exists an infinite dimensional word such that the set {f ((a n ) n ) : (a n ) n Λ ω } is monochromatic.

22 Hales-Jewett Theorem for Trees We fix a vector tree T. Fix a finite alphabet Λ. For m < n < ω, set W(Λ, T, m, n) = Λ T [m,n), where T [m, n) = n j=m T(j). We also set W(Λ, T) = m n W(Λ, T, m, n).

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24 Let (v s ) s T be a collection of distinct variables, set of symbols disjoint from Λ. Fix a vector level subset D of T. Let to be the set of all functions such that W v (Λ, T, D, m, n) f : T [m, n) Λ {v s : s D} The set f ({v s }) is nonempty and admits s as a minimum in T, for all s D. For every s and s in D, we have L T (f ({v s })) = L T (f ({v s })).

25 v s v s v s v s v s s v s v s3vs3 s v s3 v s4 s 3 s 4 v s4 v s5 v s5 s 5

26 For f W v (Λ, T, D, m, n), set Moreover, we set ws(f ) = D, bot(f ) = m and top(f ) = n. W v (Λ, T) = { Wv (Λ, T, D, m, n) : m n and D is a vector level subset of T with L T (D) [m, n) }. The elements of W v (Λ, T) are viewed as variable words over the alphabet Λ.

27 For variable words f in W v (Λ, T) we take substitutions: For every family a = (a s ) s ws(f ) Λ, let f (a) W(Λ, T) be the result of substituting for every s in ws(f ) each occurrence of v s by a s,. Moreover, we set the constant span of f. [f ] Λ = {f (a) : a = (a s ) s ws(f ) Λ},

28 An infinite sequence X = (f n ) n<ω in W v (Λ, T) is a subspace, if: bot(f 0 ) = 0. bot(f n+ ) = top(f n ) for all n < ω. 3 Setting D i = n<ω ws i(f n ) for all i =,..., d, where ws(f n ) = (ws (f n ),..., ws d (f n )), we have that (D,..., D d ) forms a dense vector subset of T. For a subspace X = (f n ) n<ω we define { n [X] Λ = q=0 } g q : n < ω and g q [f q ] Λ for all q = 0,..., n. For two subspaces X and Y, we write X Y if [X] Λ [Y] Λ.

29 An infinite sequence X = (f n ) n<ω in W v (Λ, T) is a subspace, if: bot(f 0 ) = 0. bot(f n+ ) = top(f n ) for all n < ω. 3 Setting D i = n<ω ws i(f n ) for all i =,..., d, where ws(f n ) = (ws (f n ),..., ws d (f n )), we have that (D,..., D d ) forms a dense vector subset of T. For a subspace X = (f n ) n<ω we define { n [X] Λ = q=0 } g q : n < ω and g q [f q ] Λ for all q = 0,..., n. For two subspaces X and Y, we write X Y if [X] Λ [Y] Λ.

30 An infinite Hales-Jewett theorem for trees Theorem Let Λ be a finite alphabet and T a vector tree. Then for every finite coloring of the set of the constant words W(Λ, T) over Λ and every subspace X of W(Λ, T) there exists a subspace X of W(Λ, T) with X X such that the set [X ] Λ is monochromatic.

31 A Ramsey space of sequences of words Let W (Λ, T), be the set of all sequences (g n ) n<ω in W(Λ, T) such that: bot(g 0 ) = 0 and bot(g n+ ) = topg n for all n < ω. For a subspace X, we set [X] Λ = {(g n ) n<ω W (Λ, T) : ( n < ω) n g q [X] Λ. Theorem Let Λ be a finite alphabet and T a vector tree. Then for every finite Souslin measurable coloring of the set W (Λ, T) and every subspace X of W(Λ, T) there exists a subspace X of W(Λ, T) with X X such that the set [X ] Λ is monochromatic. q=0