An application to Tukey theory For directed sets D and E , put D ≤ T E if there is a cofinal map f : E → D , i.e., a map such that ( ∀ X ⊆ E )[ X cofinal in E ⇒ f [ X ] cofinal in D ] . Let D ≡ T E whenever D ≤ T E and E ≤ T D . Theorem (Tukey, 1940) D ≡ T E holds for directed sets D and E if and only if they are isomorphic to cofinal subsets of a single directed F . We give application to Tukey classification of ultrafilters on N and its relation to the Rudin-Keisler classification . Theorem The following are equivalent for a selective ultrafilter V and an arbitrary nonprincipal ultrafilter U on N : 1. U ≤ T V .
An application to Tukey theory For directed sets D and E , put D ≤ T E if there is a cofinal map f : E → D , i.e., a map such that ( ∀ X ⊆ E )[ X cofinal in E ⇒ f [ X ] cofinal in D ] . Let D ≡ T E whenever D ≤ T E and E ≤ T D . Theorem (Tukey, 1940) D ≡ T E holds for directed sets D and E if and only if they are isomorphic to cofinal subsets of a single directed F . We give application to Tukey classification of ultrafilters on N and its relation to the Rudin-Keisler classification . Theorem The following are equivalent for a selective ultrafilter V and an arbitrary nonprincipal ultrafilter U on N : 1. U ≤ T V . 2. U ≡ RK V α for some countable ordinal α.
Here V α denotes the α th Fubini power of V defined recursively on α up to RK -equivalence in the natural way: A ∈ V α iff { i : { j : 2 i (2 j + 1) ∈ A } ∈ V α i } ∈ V where α i = α − 1 for all i when α is successor or α i ↑ α when α is a limit ordinal.
Here V α denotes the α th Fubini power of V defined recursively on α up to RK -equivalence in the natural way: A ∈ V α iff { i : { j : 2 i (2 j + 1) ∈ A } ∈ V α i } ∈ V where α i = α − 1 for all i when α is successor or α i ↑ α when α is a limit ordinal. Lemma V ≡ T V α for every selective ultrafilter V and every countable ordinal α > 0 .
Here V α denotes the α th Fubini power of V defined recursively on α up to RK -equivalence in the natural way: A ∈ V α iff { i : { j : 2 i (2 j + 1) ∈ A } ∈ V α i } ∈ V where α i = α − 1 for all i when α is successor or α i ↑ α when α is a limit ordinal. Lemma V ≡ T V α for every selective ultrafilter V and every countable ordinal α > 0 . Corollary The following are equivalent for two selective ultrafilters U and V : 1. U ≤ T V . 2. U ≤ RK V .
Here V α denotes the α th Fubini power of V defined recursively on α up to RK -equivalence in the natural way: A ∈ V α iff { i : { j : 2 i (2 j + 1) ∈ A } ∈ V α i } ∈ V where α i = α − 1 for all i when α is successor or α i ↑ α when α is a limit ordinal. Lemma V ≡ T V α for every selective ultrafilter V and every countable ordinal α > 0 . Corollary The following are equivalent for two selective ultrafilters U and V : 1. U ≤ T V . 2. U ≤ RK V . Corollary Selective ultrafilters realize minimal cofinal types in β N \ N .
Ultrafilters on barriers For a barrier B and n ∈ N , set B { n } = { s \ { n } : s ∈ B , n = min ( s ) } . Then B { n } is a barrier on N \ { 0 , 1 ...., n } for all n ∈ N .
Ultrafilters on barriers For a barrier B and n ∈ N , set B { n } = { s \ { n } : s ∈ B , n = min ( s ) } . Then B { n } is a barrier on N \ { 0 , 1 ...., n } for all n ∈ N . Fix a nonprincipal ultrafilter U on N . Define an ultrafilter U B on B as follows: X ∈ U B ⇔ ( U n ) X { n } ∈ U B { n } .
Ultrafilters on barriers For a barrier B and n ∈ N , set B { n } = { s \ { n } : s ∈ B , n = min ( s ) } . Then B { n } is a barrier on N \ { 0 , 1 ...., n } for all n ∈ N . Fix a nonprincipal ultrafilter U on N . Define an ultrafilter U B on B as follows: X ∈ U B ⇔ ( U n ) X { n } ∈ U B { n } . Example If B = [ N ] k for some positive integer k then U k = U [ N ] k is the usual k th Fubini power of U .
Ultrafilters on barriers For a barrier B and n ∈ N , set B { n } = { s \ { n } : s ∈ B , n = min ( s ) } . Then B { n } is a barrier on N \ { 0 , 1 ...., n } for all n ∈ N . Fix a nonprincipal ultrafilter U on N . Define an ultrafilter U B on B as follows: X ∈ U B ⇔ ( U n ) X { n } ∈ U B { n } . Example If B = [ N ] k for some positive integer k then U k = U [ N ] k is the usual k th Fubini power of U . The ultrafilters of the form U B for B a barrier on N will be called the countable Fubini powers of the ultrafilter U .
Suppose U ≤ T V and that V is selective. Fix a cofinal map f : V → U .
Suppose U ≤ T V and that V is selective. Fix a cofinal map f : V → U . Step 1: Show that f can be assumed to be continuous or more precisely that f = h ↾ V for some continuous h : N [ ∞ ] → 2 N .
Suppose U ≤ T V and that V is selective. Fix a cofinal map f : V → U . Step 1: Show that f can be assumed to be continuous or more precisely that f = h ↾ V for some continuous h : N [ ∞ ] → 2 N . Step 2: Consider the corresponding h 1 : N [ ∞ ] → N defined by h 1 ( M ) = min( h ( M )) .
Suppose U ≤ T V and that V is selective. Fix a cofinal map f : V → U . Step 1: Show that f can be assumed to be continuous or more precisely that f = h ↾ V for some continuous h : N [ ∞ ] → 2 N . Step 2: Consider the corresponding h 1 : N [ ∞ ] → N defined by h 1 ( M ) = min( h ( M )) . Then there is a barrier B on N and g : B → N such that h 1 ( M ) = g ( t M ) , where t M is the unique t ∈ B such that t ⊑ M .
Suppose U ≤ T V and that V is selective. Fix a cofinal map f : V → U . Step 1: Show that f can be assumed to be continuous or more precisely that f = h ↾ V for some continuous h : N [ ∞ ] → 2 N . Step 2: Consider the corresponding h 1 : N [ ∞ ] → N defined by h 1 ( M ) = min( h ( M )) . Then there is a barrier B on N and g : B → N such that h 1 ( M ) = g ( t M ) , where t M is the unique t ∈ B such that t ⊑ M . Step 3: Find M ∈ V and irreducible map ϕ : B → N such that E g ↾ ( B ↾ M ) = E ϕ ↾ ( B ↾ M ) .
Suppose U ≤ T V and that V is selective. Fix a cofinal map f : V → U . Step 1: Show that f can be assumed to be continuous or more precisely that f = h ↾ V for some continuous h : N [ ∞ ] → 2 N . Step 2: Consider the corresponding h 1 : N [ ∞ ] → N defined by h 1 ( M ) = min( h ( M )) . Then there is a barrier B on N and g : B → N such that h 1 ( M ) = g ( t M ) , where t M is the unique t ∈ B such that t ⊑ M . Step 3: Find M ∈ V and irreducible map ϕ : B → N such that E g ↾ ( B ↾ M ) = E ϕ ↾ ( B ↾ M ) . Step 4: Show that this means that U ≡ RK V ϕ [ B ] .
Topological Ramsey spaces A topological Ramsey space is a set R of sequences A = ( a k ) of objects and a quasi-ordering ≤ which defines the corresponding topology of basic-open sets of the form [ n , B ] = { A ∈ R : A ≤ B and a k = b k for all n < k } with the property that all Baire subsets X of R are Ramsey , i.e., ( ∀ n < ω )( ∀ B ∈ R )( ∃ A ∈ [ n , B ])[[ n , A ] ⊆ X or [ n , A ] ∩ X = ∅ ] .
Topological Ramsey spaces A topological Ramsey space is a set R of sequences A = ( a k ) of objects and a quasi-ordering ≤ which defines the corresponding topology of basic-open sets of the form [ n , B ] = { A ∈ R : A ≤ B and a k = b k for all n < k } with the property that all Baire subsets X of R are Ramsey , i.e., ( ∀ n < ω )( ∀ B ∈ R )( ∃ A ∈ [ n , B ])[[ n , A ] ⊆ X or [ n , A ] ∩ X = ∅ ] . Notation: r n ( A ) = � a k : k < n � is the n th approximation to A . AR is the collection of all approximations to sequences in R . | a | is the length of an approximation a ∈ AR . AR l = { a ∈ AR : | a | = l } . [ a , B ] = { A ∈ R : A ≤ B and r | a | ( A ) = a } .
A sufficient condition for R being a topological Ramsey space is that R is a closed subset of AR ω . and that the triple ( R , ≤ , r ) has the following properties:
A sufficient condition for R being a topological Ramsey space is that R is a closed subset of AR ω . and that the triple ( R , ≤ , r ) has the following properties: A.1. Finitization There is a quasi-ordering ≤ fin on AR such that (1) { a ∈ AR : a ≤ fin b } is finite for all b ∈ AR , (2) A ≤ B iff ( ∀ n )( ∃ m ) r n ( A ) ≤ fin r m ( B ) , (3) ∀ a , b ∈ AR [ a ⊑ b ∧ b ≤ fin c → ∃ d ⊑ c a ≤ fin d ] .
A sufficient condition for R being a topological Ramsey space is that R is a closed subset of AR ω . and that the triple ( R , ≤ , r ) has the following properties: A.1. Finitization There is a quasi-ordering ≤ fin on AR such that (1) { a ∈ AR : a ≤ fin b } is finite for all b ∈ AR , (2) A ≤ B iff ( ∀ n )( ∃ m ) r n ( A ) ≤ fin r m ( B ) , (3) ∀ a , b ∈ AR [ a ⊑ b ∧ b ≤ fin c → ∃ d ⊑ c a ≤ fin d ] . A.2. Amalgamation (1) If depth B ( a ) < ∞ then [ a , A ] � = ∅ for all A ∈ [ depth B ( a ) , B ]. (2) A ≤ B and [ a , A ] � = ∅ imply that there is A ′ ∈ [ depth B ( a ) , B ] such that ∅ � = [ a , A ′ ] ⊆ [ a , A ].
A sufficient condition for R being a topological Ramsey space is that R is a closed subset of AR ω . and that the triple ( R , ≤ , r ) has the following properties: A.1. Finitization There is a quasi-ordering ≤ fin on AR such that (1) { a ∈ AR : a ≤ fin b } is finite for all b ∈ AR , (2) A ≤ B iff ( ∀ n )( ∃ m ) r n ( A ) ≤ fin r m ( B ) , (3) ∀ a , b ∈ AR [ a ⊑ b ∧ b ≤ fin c → ∃ d ⊑ c a ≤ fin d ] . A.2. Amalgamation (1) If depth B ( a ) < ∞ then [ a , A ] � = ∅ for all A ∈ [ depth B ( a ) , B ]. (2) A ≤ B and [ a , A ] � = ∅ imply that there is A ′ ∈ [ depth B ( a ) , B ] such that ∅ � = [ a , A ′ ] ⊆ [ a , A ]. A.3. Pigeon-Hole If depth B ( a ) < ∞ and if O ⊆ AR | a | +1 , then there is A ∈ [ depth B ( a ) , B ] such that r | a | +1 [ a , A ] ⊆ O or r | a | +1 [ a , A ] ⊆ O c .
The Ramsey-Galvin-Prikry-Ellentuck space Let R = N [ ∞ ] = { A ⊆ N : | A | = ℵ 0 } , ≤ = ⊆ , and r n ( A ) = { first n members of A } .
The Ramsey-Galvin-Prikry-Ellentuck space Let R = N [ ∞ ] = { A ⊆ N : | A | = ℵ 0 } , ≤ = ⊆ , and r n ( A ) = { first n members of A } . Theorem (Ellentuck, 1974) N [ ∞ ] is a topological Ramsey space.
The Ramsey-Galvin-Prikry-Ellentuck space Let R = N [ ∞ ] = { A ⊆ N : | A | = ℵ 0 } , ≤ = ⊆ , and r n ( A ) = { first n members of A } . Theorem (Ellentuck, 1974) N [ ∞ ] is a topological Ramsey space. Theorem (Galvin-Prikry, 1973) For every finite Borel coloring of N [ ∞ ] there is infinite M ⊆ N such that M [ ∞ ] is monochromatic.
The Ramsey-Galvin-Prikry-Ellentuck space Let R = N [ ∞ ] = { A ⊆ N : | A | = ℵ 0 } , ≤ = ⊆ , and r n ( A ) = { first n members of A } . Theorem (Ellentuck, 1974) N [ ∞ ] is a topological Ramsey space. Theorem (Galvin-Prikry, 1973) For every finite Borel coloring of N [ ∞ ] there is infinite M ⊆ N such that M [ ∞ ] is monochromatic. Theorem (Erd˝ os-Rado, 1950) For every positive integer k and every equivalence relation ∼ on N [ k ] there is I ⊆ k and infinite M ⊆ N such that for a , b ∈ N [ k ] , a ∼ b iff ( ∀ i ∈ I ) a i = b i .
The Ramsey-Galvin-Prikry-Ellentuck space Let R = N [ ∞ ] = { A ⊆ N : | A | = ℵ 0 } , ≤ = ⊆ , and r n ( A ) = { first n members of A } . Theorem (Ellentuck, 1974) N [ ∞ ] is a topological Ramsey space. Theorem (Galvin-Prikry, 1973) For every finite Borel coloring of N [ ∞ ] there is infinite M ⊆ N such that M [ ∞ ] is monochromatic. Theorem (Erd˝ os-Rado, 1950) For every positive integer k and every equivalence relation ∼ on N [ k ] there is I ⊆ k and infinite M ⊆ N such that for a , b ∈ N [ k ] , a ∼ b iff ( ∀ i ∈ I ) a i = b i . Theorem (Ramsey, 1930) For every positive integer k and every finite coloring of N [ k ] there is infinite M ⊆ N such that M [ k ] is monochromatic.
The Halpern-L¨ auchli space of strong subtrees
The Halpern-L¨ auchli space of strong subtrees Let U be a fixed rooted finitely branching tree of height ω with no terminal nodes and we study its subtrees. We assume U ⊆ ω <ω .
The Halpern-L¨ auchli space of strong subtrees Let U be a fixed rooted finitely branching tree of height ω with no terminal nodes and we study its subtrees. We assume U ⊆ ω <ω . We say that T is a strong subtree of U if T is also rooted and if there is A = { n i } ⊆ ω, the level-set of T , such that
The Halpern-L¨ auchli space of strong subtrees Let U be a fixed rooted finitely branching tree of height ω with no terminal nodes and we study its subtrees. We assume U ⊆ ω <ω . We say that T is a strong subtree of U if T is also rooted and if there is A = { n i } ⊆ ω, the level-set of T , such that (1) the i th level T ( i ) of the tree T is a subset of the n i th level U ( n i ) of the tree U ,
The Halpern-L¨ auchli space of strong subtrees Let U be a fixed rooted finitely branching tree of height ω with no terminal nodes and we study its subtrees. We assume U ⊆ ω <ω . We say that T is a strong subtree of U if T is also rooted and if there is A = { n i } ⊆ ω, the level-set of T , such that (1) the i th level T ( i ) of the tree T is a subset of the n i th level U ( n i ) of the tree U , (2) if m < n are two successive elements of the set A , then for every s ∈ T ∩ U ( m ) , every immediate successor of s in U has exactly one extension in T ∩ U ( n ) .
The Halpern-L¨ auchli space of strong subtrees Let U be a fixed rooted finitely branching tree of height ω with no terminal nodes and we study its subtrees. We assume U ⊆ ω <ω . We say that T is a strong subtree of U if T is also rooted and if there is A = { n i } ⊆ ω, the level-set of T , such that (1) the i th level T ( i ) of the tree T is a subset of the n i th level U ( n i ) of the tree U , (2) if m < n are two successive elements of the set A , then for every s ∈ T ∩ U ( m ) , every immediate successor of s in U has exactly one extension in T ∩ U ( n ) . S k ( U ) = the collection of all strong subtrees of U of height k .
The Halpern-L¨ auchli space of strong subtrees Let U be a fixed rooted finitely branching tree of height ω with no terminal nodes and we study its subtrees. We assume U ⊆ ω <ω . We say that T is a strong subtree of U if T is also rooted and if there is A = { n i } ⊆ ω, the level-set of T , such that (1) the i th level T ( i ) of the tree T is a subset of the n i th level U ( n i ) of the tree U , (2) if m < n are two successive elements of the set A , then for every s ∈ T ∩ U ( m ) , every immediate successor of s in U has exactly one extension in T ∩ U ( n ) . S k ( U ) = the collection of all strong subtrees of U of height k . Theorem (Milliken, 1981) S ω ( U ) is a topological Ramsey space.
The Halpern-L¨ auchli space of strong subtrees Let U be a fixed rooted finitely branching tree of height ω with no terminal nodes and we study its subtrees. We assume U ⊆ ω <ω . We say that T is a strong subtree of U if T is also rooted and if there is A = { n i } ⊆ ω, the level-set of T , such that (1) the i th level T ( i ) of the tree T is a subset of the n i th level U ( n i ) of the tree U , (2) if m < n are two successive elements of the set A , then for every s ∈ T ∩ U ( m ) , every immediate successor of s in U has exactly one extension in T ∩ U ( n ) . S k ( U ) = the collection of all strong subtrees of U of height k . Theorem (Milliken, 1981) S ω ( U ) is a topological Ramsey space. Corollary For every finite Borel coloring of S ω ( U ) there is a strong subtree T of U of height ω such that S ω ( T ) is monochromatic.
Corollary For every positive integer k and every finite coloring of S k ( U ) there is T ∈ S ω ( U ) such that S k ( T ) is monochromatic.
Corollary For every positive integer k and every finite coloring of S k ( U ) there is T ∈ S ω ( U ) such that S k ( T ) is monochromatic. Corollary For every equivalence relation ∼ on U there is a strong subtree T of U of height ω such that one of the following holds: 1. ( ∀ s , t ∈ T )[ s ∼ t ⇔ s = t ] , 2. ( ∀ s , t ∈ T )[ s ∼ t ⇔ s = s ] , 3. ( ∀ s , t ∈ T )[ s ∼ t ⇔ | s | = | t | ] ,
Corollary For every positive integer k and every finite coloring of S k ( U ) there is T ∈ S ω ( U ) such that S k ( T ) is monochromatic. Corollary For every equivalence relation ∼ on U there is a strong subtree T of U of height ω such that one of the following holds: 1. ( ∀ s , t ∈ T )[ s ∼ t ⇔ s = t ] , 2. ( ∀ s , t ∈ T )[ s ∼ t ⇔ s = s ] , 3. ( ∀ s , t ∈ T )[ s ∼ t ⇔ | s | = | t | ] , Corollary U contains a strong subtree S of height ω such that either 1. S is uniformly branching of some degree d 2. nodes of S of the same height have the same degree which increase as the height increase 3. different nodes of S have different branching degrees.
Fix a positive integer d and consider the tree U d = d <ω with the usual end-extension ordering ⊑ and the lexicographical ordering.
Fix a positive integer d and consider the tree U d = d <ω with the usual end-extension ordering ⊑ and the lexicographical ordering. Then every pair of strong subtrees A and B of U of some fixed height k are isomorphic as they are isomorphic to d < k . Note that there is a unique isomorphism preserving the lexicographical ordering. We shall use only about this kind of isomorphisms.
Fix a positive integer d and consider the tree U d = d <ω with the usual end-extension ordering ⊑ and the lexicographical ordering. Then every pair of strong subtrees A and B of U of some fixed height k are isomorphic as they are isomorphic to d < k . Note that there is a unique isomorphism preserving the lexicographical ordering. We shall use only about this kind of isomorphisms. Theorem (Milliken, 1980) For every positive integer k and every equivalence relation ∼ on S k ( U d ) there is a strong subtree T of U d and a pair ( N , L ) ∈ P ( d < k ) × P ( k ) with max {| s | : s ∈ N } < min( L ) such that A , B ∈ S k ( T ) , we have that A ∼ B iff
Fix a positive integer d and consider the tree U d = d <ω with the usual end-extension ordering ⊑ and the lexicographical ordering. Then every pair of strong subtrees A and B of U of some fixed height k are isomorphic as they are isomorphic to d < k . Note that there is a unique isomorphism preserving the lexicographical ordering. We shall use only about this kind of isomorphisms. Theorem (Milliken, 1980) For every positive integer k and every equivalence relation ∼ on S k ( U d ) there is a strong subtree T of U d and a pair ( N , L ) ∈ P ( d < k ) × P ( k ) with max {| s | : s ∈ N } < min( L ) such that A , B ∈ S k ( T ) , we have that A ∼ B iff (1) the isomorphisms between d < k and A and B agree on N ,
Fix a positive integer d and consider the tree U d = d <ω with the usual end-extension ordering ⊑ and the lexicographical ordering. Then every pair of strong subtrees A and B of U of some fixed height k are isomorphic as they are isomorphic to d < k . Note that there is a unique isomorphism preserving the lexicographical ordering. We shall use only about this kind of isomorphisms. Theorem (Milliken, 1980) For every positive integer k and every equivalence relation ∼ on S k ( U d ) there is a strong subtree T of U d and a pair ( N , L ) ∈ P ( d < k ) × P ( k ) with max {| s | : s ∈ N } < min( L ) such that A , B ∈ S k ( T ) , we have that A ∼ B iff (1) the isomorphisms between d < k and A and B agree on N , (2) for l ∈ L , the l-th levels of A and B are subsets of the same level of T .
The countable dense linear ordering
The countable dense linear ordering Theorem (Devlin-Laver 1979) For every positive integer k and every finite coloring of the set [ Q ] k of all k-element subsets of Q there is P ⊆ Q order-isomorphic to Q such that [ P ] k uses at most t k colors. Here, t k = � k − 1 � 2 k − 2 � t l · t k − l is the standard sequence of odd l =1 2 l − 1 tangent numbers: t 1 = 1 , t 2 = 2 , t 3 = 16 , etc.
The countable dense linear ordering Theorem (Devlin-Laver 1979) For every positive integer k and every finite coloring of the set [ Q ] k of all k-element subsets of Q there is P ⊆ Q order-isomorphic to Q such that [ P ] k uses at most t k colors. Here, t k = � k − 1 � 2 k − 2 � t l · t k − l is the standard sequence of odd l =1 2 l − 1 tangent numbers: t 1 = 1 , t 2 = 2 , t 3 = 16 , etc. We apply the strong-subtree Ramsey theorem for U = 2 <ω and after getting monochromatic T ∈ S ω ( U ) we consider the subtree that corresponds to the following: Let S be the ∧ -closed subtree of 2 <ω uniquely determined by the following properties:
The countable dense linear ordering Theorem (Devlin-Laver 1979) For every positive integer k and every finite coloring of the set [ Q ] k of all k-element subsets of Q there is P ⊆ Q order-isomorphic to Q such that [ P ] k uses at most t k colors. Here, t k = � k − 1 � 2 k − 2 � t l · t k − l is the standard sequence of odd l =1 2 l − 1 tangent numbers: t 1 = 1 , t 2 = 2 , t 3 = 16 , etc. We apply the strong-subtree Ramsey theorem for U = 2 <ω and after getting monochromatic T ∈ S ω ( U ) we consider the subtree that corresponds to the following: Let S be the ∧ -closed subtree of 2 <ω uniquely determined by the following properties: (1) root ( S ) = ∅ and S is isomorphic to 2 <ω ,
The countable dense linear ordering Theorem (Devlin-Laver 1979) For every positive integer k and every finite coloring of the set [ Q ] k of all k-element subsets of Q there is P ⊆ Q order-isomorphic to Q such that [ P ] k uses at most t k colors. Here, t k = � k − 1 � 2 k − 2 � t l · t k − l is the standard sequence of odd l =1 2 l − 1 tangent numbers: t 1 = 1 , t 2 = 2 , t 3 = 16 , etc. We apply the strong-subtree Ramsey theorem for U = 2 <ω and after getting monochromatic T ∈ S ω ( U ) we consider the subtree that corresponds to the following: Let S be the ∧ -closed subtree of 2 <ω uniquely determined by the following properties: (1) root ( S ) = ∅ and S is isomorphic to 2 <ω , (2) | S ∩ 2 3 n | = 1 and S ∩ 2 3 n +1 = S ∩ 2 3 n +2 = ∅ for all n ,
The countable dense linear ordering Theorem (Devlin-Laver 1979) For every positive integer k and every finite coloring of the set [ Q ] k of all k-element subsets of Q there is P ⊆ Q order-isomorphic to Q such that [ P ] k uses at most t k colors. Here, t k = � k − 1 � 2 k − 2 � t l · t k − l is the standard sequence of odd l =1 2 l − 1 tangent numbers: t 1 = 1 , t 2 = 2 , t 3 = 16 , etc. We apply the strong-subtree Ramsey theorem for U = 2 <ω and after getting monochromatic T ∈ S ω ( U ) we consider the subtree that corresponds to the following: Let S be the ∧ -closed subtree of 2 <ω uniquely determined by the following properties: (1) root ( S ) = ∅ and S is isomorphic to 2 <ω , (2) | S ∩ 2 3 n | = 1 and S ∩ 2 3 n +1 = S ∩ 2 3 n +2 = ∅ for all n , (3) ( ∀ m )( ∀ s , t ∈ S ( m ))( s < lex t ⇒ | s | < | t | ),
The countable dense linear ordering Theorem (Devlin-Laver 1979) For every positive integer k and every finite coloring of the set [ Q ] k of all k-element subsets of Q there is P ⊆ Q order-isomorphic to Q such that [ P ] k uses at most t k colors. Here, t k = � k − 1 � 2 k − 2 � t l · t k − l is the standard sequence of odd l =1 2 l − 1 tangent numbers: t 1 = 1 , t 2 = 2 , t 3 = 16 , etc. We apply the strong-subtree Ramsey theorem for U = 2 <ω and after getting monochromatic T ∈ S ω ( U ) we consider the subtree that corresponds to the following: Let S be the ∧ -closed subtree of 2 <ω uniquely determined by the following properties: (1) root ( S ) = ∅ and S is isomorphic to 2 <ω , (2) | S ∩ 2 3 n | = 1 and S ∩ 2 3 n +1 = S ∩ 2 3 n +2 = ∅ for all n , (3) ( ∀ m )( ∀ s , t ∈ S ( m ))( s < lex t ⇒ | s | < | t | ), (4) ( ∀ m < n )( ∀ s ∈ S ( m ))( ∀ t ∈ S ( n )) | s | < | t | ,
The countable dense linear ordering Theorem (Devlin-Laver 1979) For every positive integer k and every finite coloring of the set [ Q ] k of all k-element subsets of Q there is P ⊆ Q order-isomorphic to Q such that [ P ] k uses at most t k colors. Here, t k = � k − 1 � 2 k − 2 � t l · t k − l is the standard sequence of odd l =1 2 l − 1 tangent numbers: t 1 = 1 , t 2 = 2 , t 3 = 16 , etc. We apply the strong-subtree Ramsey theorem for U = 2 <ω and after getting monochromatic T ∈ S ω ( U ) we consider the subtree that corresponds to the following: Let S be the ∧ -closed subtree of 2 <ω uniquely determined by the following properties: (1) root ( S ) = ∅ and S is isomorphic to 2 <ω , (2) | S ∩ 2 3 n | = 1 and S ∩ 2 3 n +1 = S ∩ 2 3 n +2 = ∅ for all n , (3) ( ∀ m )( ∀ s , t ∈ S ( m ))( s < lex t ⇒ | s | < | t | ), (4) ( ∀ m < n )( ∀ s ∈ S ( m ))( ∀ t ∈ S ( n )) | s | < | t | , (5) ( ∀ s ∈ S )( ∀ t �∈ S )( t ⊑ s ⇒ t ⌢ (0) ⊑ s ) .
Ramsey-classification problem for [ Q ] k
Ramsey-classification problem for [ Q ] k Theorem (Vuksanovic, 2003) For every positive integer k the class of all equivalence relations on the set [ Q ] k of all k-element subsets of Q has a Ramsey basis of equivalence relations E T determined by ’transitive sets’ T ⊆ [2 ≤ 4 k − 2 ] × [2 ≤ 4 k − 2 ] . In case k = 1 the Ramsey basis has 2 elements and in case k = 2 it has 57 elements.
Ramsey-classification problem for [ Q ] k Theorem (Vuksanovic, 2003) For every positive integer k the class of all equivalence relations on the set [ Q ] k of all k-element subsets of Q has a Ramsey basis of equivalence relations E T determined by ’transitive sets’ T ⊆ [2 ≤ 4 k − 2 ] × [2 ≤ 4 k − 2 ] . In case k = 1 the Ramsey basis has 2 elements and in case k = 2 it has 57 elements. Remark The exact size of an irredundant Ramsey basis of the class of equivalence relations on [ Q ] 3 is not known.
Theorem The set Q [ ∞ ] of infinite of rapidly increasing sequences of rational numbers is a Ramsey space.
Theorem The set Q [ ∞ ] of infinite of rapidly increasing sequences of rational numbers is a Ramsey space. Corollary For every finite Borel coloring of Q [ ∞ ] there is P ⊆ Q order-isomorphic to Q such that P [ ∞ ] is monochromatic.
Theorem The set Q [ ∞ ] of infinite of rapidly increasing sequences of rational numbers is a Ramsey space. Corollary For every finite Borel coloring of Q [ ∞ ] there is P ⊆ Q order-isomorphic to Q such that P [ ∞ ] is monochromatic. Corollary For every positive integer k and every finite coloring of the set Q [ k ] of rapidly increasing k-sequences of elements of Q there is P ⊆ Q order-isomorphic to Q such that P [ k ] is monochromatic.
Theorem The set Q [ ∞ ] of infinite of rapidly increasing sequences of rational numbers is a Ramsey space. Corollary For every finite Borel coloring of Q [ ∞ ] there is P ⊆ Q order-isomorphic to Q such that P [ ∞ ] is monochromatic. Corollary For every positive integer k and every finite coloring of the set Q [ k ] of rapidly increasing k-sequences of elements of Q there is P ⊆ Q order-isomorphic to Q such that P [ k ] is monochromatic. Theorem For every positive integer k the collection of equivalence relations on Q [ k ] has a finite Ramsey basis of cardinality h k = 2 � k i =0 5 i .
The Hindman-Milliken-Taylor space FIN [ ∞ ]
The Hindman-Milliken-Taylor space FIN [ ∞ ] We work with the the countable dense-in-itself metric space Q = {∅} ∪ FIN ⊆ 2 N , where FIN is the collection of all finite nonempty subsets of N .
The Hindman-Milliken-Taylor space FIN [ ∞ ] We work with the the countable dense-in-itself metric space Q = {∅} ∪ FIN ⊆ 2 N , where FIN is the collection of all finite nonempty subsets of N . Theorem (Milliken, 1975) The collection FIN [ ∞ ] of all infinite sequences of elements of FIN converging to ∅ is a topological Ramsey space.
The Hindman-Milliken-Taylor space FIN [ ∞ ] We work with the the countable dense-in-itself metric space Q = {∅} ∪ FIN ⊆ 2 N , where FIN is the collection of all finite nonempty subsets of N . Theorem (Milliken, 1975) The collection FIN [ ∞ ] of all infinite sequences of elements of FIN converging to ∅ is a topological Ramsey space. If X = ( x n ) ∈ FIN [ ∞ ] then max( x m ) < min( x n ) whenever m < n and for X = ( x n ) , Y = ( y n ) ∈ FIN [ ∞ ] , we set X ≤ Y whenever X ⊆ [ Y ] , where [ Y ] = { y n 0 ∪ · · · y n k : n 0 < · · · < n k } .
Corollary For every finite Borel coloring of FIN [ ∞ ] there is Y = ( y n ) ∈ FIN [ ∞ ] such that [ Y ] [ ∞ ] is monochromatic.
Corollary For every finite Borel coloring of FIN [ ∞ ] there is Y = ( y n ) ∈ FIN [ ∞ ] such that [ Y ] [ ∞ ] is monochromatic. Corollary For every positive integer k and every finite coloring of FIN [ k ] there is Y = ( y n ) ∈ FIN [ ∞ ] such that [ Y ] [ k ] is monochromatic.
Corollary For every finite Borel coloring of FIN [ ∞ ] there is Y = ( y n ) ∈ FIN [ ∞ ] such that [ Y ] [ ∞ ] is monochromatic. Corollary For every positive integer k and every finite coloring of FIN [ k ] there is Y = ( y n ) ∈ FIN [ ∞ ] such that [ Y ] [ k ] is monochromatic. Theorem (Taylor, 1976) For every equivalence relation E on FIN there is Y = ( y n ) ∈ FIN [ ∞ ] and ϕ ∈ { const , ident , min , max , (min , max) } such that E ↾ [ Y ] = E ϕ ↾ [ Y ] .
Corollary For every finite Borel coloring of FIN [ ∞ ] there is Y = ( y n ) ∈ FIN [ ∞ ] such that [ Y ] [ ∞ ] is monochromatic. Corollary For every positive integer k and every finite coloring of FIN [ k ] there is Y = ( y n ) ∈ FIN [ ∞ ] such that [ Y ] [ k ] is monochromatic. Theorem (Taylor, 1976) For every equivalence relation E on FIN there is Y = ( y n ) ∈ FIN [ ∞ ] and ϕ ∈ { const , ident , min , max , (min , max) } such that E ↾ [ Y ] = E ϕ ↾ [ Y ] . Theorem (Lefmann, 1996) For every positive integer k the class of equivalence relations on FIN [ k ] has a Ramsey basis of cardinality √ √ √ √ 13) k + (13 − 3 1 13) k ] . 13)(7 − s k = 13 · 2 k +1 [(13 + 3 13)(7 +
Ramsey on countable topological spaces
Ramsey on countable topological spaces Theorem (Baumgartner, 1986) Suppose X is a countable Hausdorff topological space. Then there is c : [ X ] 2 → ω such that c [ P ] 2 ⊇ { 0 , 1 , ..., 2 n − 1 } for all n < ω and all P ⊆ X such that P ( n ) � = ∅ Here P (0) = P , P ( n +1) = ( P ( n ) ) ′ , where for a subset A of X we let A ′ = { x ∈ A : x ∈ A \ { x }} .
Ramsey on countable topological spaces Theorem (Baumgartner, 1986) Suppose X is a countable Hausdorff topological space. Then there is c : [ X ] 2 → ω such that c [ P ] 2 ⊇ { 0 , 1 , ..., 2 n − 1 } for all n < ω and all P ⊆ X such that P ( n ) � = ∅ Here P (0) = P , P ( n +1) = ( P ( n ) ) ′ , where for a subset A of X we let A ′ = { x ∈ A : x ∈ A \ { x }} . Theorem (Baumgartner, 1986) For every positive integer n and every finite coloring of [ ε 0 ] 2 there is P ⊆ ε 0 order-homeomorphic to ω n + 1 such that [ P ] 2 uses no more than 2 n colors.
FIN as a topological copy of Q
FIN as a topological copy of Q Recall that FIN is the space of all finite nonempty subsets of N and we shall take it with the topology induced from 2 N as our topological copy of the rationals Q .
FIN as a topological copy of Q Recall that FIN is the space of all finite nonempty subsets of N and we shall take it with the topology induced from 2 N as our topological copy of the rationals Q . For X ⊆ Q , let ∂ 0 ( X ) = X and ∂ k +1 ( X ) = ∂ ( ∂ k ( X )) , where ∂ ( X ) = { x ∈ Q : x ∈ X \ { x }} . Thus X ′ = ∂ ( X ) ∩ X so the two derivatives agree on closed subsets of Q .
FIN as a topological copy of Q Recall that FIN is the space of all finite nonempty subsets of N and we shall take it with the topology induced from 2 N as our topological copy of the rationals Q . For X ⊆ Q , let ∂ 0 ( X ) = X and ∂ k +1 ( X ) = ∂ ( ∂ k ( X )) , where ∂ ( X ) = { x ∈ Q : x ∈ X \ { x }} . Thus X ′ = ∂ ( X ) ∩ X so the two derivatives agree on closed subsets of Q . Theorem For every positive integers k and n there is integer h ( n , k ) such that for every finite coloring of [ Q ] k there is P ⊆ Q homeomorphic to ω n such that [ P ] k uses at most h ( k , n ) colors.
FIN as a topological copy of Q Recall that FIN is the space of all finite nonempty subsets of N and we shall take it with the topology induced from 2 N as our topological copy of the rationals Q . For X ⊆ Q , let ∂ 0 ( X ) = X and ∂ k +1 ( X ) = ∂ ( ∂ k ( X )) , where ∂ ( X ) = { x ∈ Q : x ∈ X \ { x }} . Thus X ′ = ∂ ( X ) ∩ X so the two derivatives agree on closed subsets of Q . Theorem For every positive integers k and n there is integer h ( n , k ) such that for every finite coloring of [ Q ] k there is P ⊆ Q homeomorphic to ω n such that [ P ] k uses at most h ( k , n ) colors. Remark The function seems expressible using the standard enumerating functions. For example, h ( n , 2) = 2 n for all n .
The oscillation mapping
The oscillation mapping The oscillation mapping on Q = {∅} ∪ FIN ⊆ 2 N : Define osc : Q 2 → ω by osc ( s , t ) = | ( s △ t ) / ∼ | , where for i , j ∈ s △ t , we let i ∼ j iff [ i , j ] ∩ ( s \ t ) = ∅ or [ i , j ] ∩ ( t \ s ) = ∅ .
The oscillation mapping The oscillation mapping on Q = {∅} ∪ FIN ⊆ 2 N : Define osc : Q 2 → ω by osc ( s , t ) = | ( s △ t ) / ∼ | , where for i , j ∈ s △ t , we let i ∼ j iff [ i , j ] ∩ ( s \ t ) = ∅ or [ i , j ] ∩ ( t \ s ) = ∅ . Proposition Suppose that ∂ k ( X ) � = ∅ for some X ⊆ Q and some positive integer k . then osc [ X 2 ] ⊇ { 2 , 3 , ..., 2 k } . If, moreover, X ∩ ∂ ( X ) � = ∅ then 1 ∈ osc [ X 2 ] as well.
Corollary The class of equivalence relations on the square of the countable dense-in-itself metric space has no finite (and, in fact, no countable) Ramsey basis.
Corollary The class of equivalence relations on the square of the countable dense-in-itself metric space has no finite (and, in fact, no countable) Ramsey basis. However, the following fact shows that the oscillation mapping is in some sense canonical . Proposition For every f : [ Q ] 2 → ω and every positive integer n there is X ⊆ Q homeomorphic to Q such that for { s , t } , { s ′ , t ′ } ∈ [ X ] 2 , osc ( s , t ) = osc ( s ′ , t ′ ) implies f ( s , t ) = f ( s ′ , t ′ ) provided that the numbers f ( s , t ) , f ( s ′ , t ′ ) , osc ( s , t ) and osc ( s ′ , t ′ ) are all ≤ n .
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