A disjoint union theorem for trees Konstantinos Tyros University of - PowerPoint PPT Presentation

A disjoint union theorem for trees Konstantinos Tyros University of Warwick Mathematics Institute Fields Institute, 2015 Konstantinos Tyros A disjoint union theorem for trees

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 = 1 of pairwise disjoint nonempty subsets of X such that the family � � � U ( D ) = D i : ∅ � = I ⊆ { 1 , 2 , ..., m } i ∈ I of non-empty unions is monochromatic. Konstantinos Tyros A disjoint union theorem for trees

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 n ∈ M is monochromatic. Konstantinos Tyros A disjoint union theorem for trees

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 . Konstantinos Tyros A disjoint union theorem for trees

Levels For n < ω , the n -th level of T , is the set T ( n ) = { t ∈ T : | Pred T ( t ) | = n } . T (3) T (2) T (1) T (0) Konstantinos Tyros A disjoint union theorem for trees

Level set For a subset D of T , we define its level set L T ( D ) = { n ∈ ω : D ∩ T ( n ) � = ∅} L T ( D ) = { 1 , 3 } Konstantinos Tyros A disjoint union theorem for trees

Vector trees From now on, fix an integer d ≥ 1. A vector tree T = ( T 1 , ..., T d ) is a d -sequence of uniquely rooted and finitely branching trees with no maximal nodes. T 1 T 2 T d Konstantinos Tyros A disjoint union theorem for trees

Level products For a vector tree T = ( T 1 , ..., T d ) we define its level product as � ⊗ T = T 1 ( n ) × ... × T d ( n ) n <ω The n -th level of the level product of T is ⊗ T ( n ) = T 1 ( n ) × ... × T d ( n ) . × → ⊗ T (3) × → ⊗ T (2) × → ⊗ T (1) × → ⊗ T (0) T 1 T 2 T d Konstantinos Tyros A disjoint union theorem for trees

Vector trees Let T = ( T 1 , ..., T d ) a vector tree. For t = ( t 1 , ..., t d ) and s = ( s 1 , ..., s d ) in ⊗ T , set t ≤ T s iff t i ≤ T i s i for all i = 1 , ..., d . For t = ( t 1 , ..., t d ) in ⊗ T , we define Succ T ( t ) = { s ∈ ⊗ T : t ≤ T s } Konstantinos Tyros A disjoint union theorem for trees

Vector subsets and dense vector subsets products A sequence D = ( D 1 , ..., D d ) is called a vector subset of T if D i is a subset of T i for all i = 1 , ..., d and L T 1 ( D 1 ) = ... = L T d ( D d ) . For a vector subset D of T we define its level product � ⊗ D = ( T 1 ( n ) ∩ D 1 ) × ... × ( 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. Konstantinos Tyros A disjoint union theorem for trees

Vector subsets and dense vector subsets products A sequence D = ( D 1 , ..., D d ) is called a vector subset of T if D i is a subset of T i for all i = 1 , ..., d and L T 1 ( D 1 ) = ... = L T d ( D d ) . For a vector subset D of T we define its level product � ⊗ D = ( T 1 ( n ) ∩ D 1 ) × ... × ( 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. Konstantinos Tyros A disjoint union theorem for trees

Dense vector subset D D D t T ( ∀ n )( ∃ m )( ∀ s ∈ ⊗ T ( n ) ∩ Succ T ( t )( ∃ s ′ ∈ ⊗ T ( m ) ∩ ⊗ D ) s ≤ T s ′ . Konstantinos Tyros A disjoint union theorem for trees

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 . Konstantinos Tyros A disjoint union theorem for trees

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 , 1 } ⊗ 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 , 1 U s ∩ U t = ∅ for s � = t , 2 min U t = t for all t ∈ ⊗ D . 3 Konstantinos Tyros A disjoint union theorem for trees

The span of a subspace For a subspace U = ( U t ) t ∈⊗ D ( U ) we define its span by � � � [ U ] = U t : Γ ⊆ ⊗ D ( U ) ∩ U ( T ) t ∈ Γ � � � = U t : Γ ⊆ ⊗ D ( U ) and Γ ∈ U ( T ) . t ∈ Γ 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 ) . Konstantinos Tyros A disjoint union theorem for trees

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 . Konstantinos Tyros A disjoint union theorem for trees

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 − 1 ( v n ) � = ∅ and max f − 1 ( v n ) < min f − 1 ( v n + 1 ) . 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. Konstantinos Tyros A disjoint union theorem for trees

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 − 1 ( v n ) � = ∅ and max f − 1 ( v n ) < min f − 1 ( v n + 1 ) . 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. Konstantinos Tyros A disjoint union theorem for trees

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 − 1 ( v n ) � = ∅ and max f − 1 ( v n ) < min f − 1 ( v n + 1 ) . 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. Konstantinos Tyros A disjoint union theorem for trees