Self-dual Ramsey theorem Definition A function s : [ L ] → [ K ] is a rigid surjection if it is surjective and images of initial segments of [ L ] are initial segments of [ K ]. s is a rigid surjection iff s ( x ) ≤ 1 + max y < x s ( y ) for each x ∈ [ L ]. Theorem (Graham–Rothschild) Given K, L and d > 0 there exists M such that for each d-coloring of all rigid surjections [ M ] → [ K ] there exists a rigid surjection t 0 : [ M ] → [ L ] such that { s ◦ t 0 : s : [ L ] → [ K ] a rigid surjection } is monochromatic. S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 7 / 42
Self-dual Ramsey theorem Definition A pair ( s , i ) is a connection between L and K if s : [ L ] → [ K ], i : [ K ] → [ L ] and for each x ∈ [ K ] s ( i ( x )) = x and ∀ y < i ( x ) s ( y ) ≤ x . S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 8 / 42
Self-dual Ramsey theorem Definition A pair ( s , i ) is a connection between L and K if s : [ L ] → [ K ], i : [ K ] → [ L ] and for each x ∈ [ K ] s ( i ( x )) = x and ∀ y < i ( x ) s ( y ) ≤ x . So, i is a left inverse of s and at each x ∈ [ K ] the value i ( x ) is picked only from those elements of s − 1 ( x ) that are “visible from x ,” S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 8 / 42
Self-dual Ramsey theorem Definition A pair ( s , i ) is a connection between L and K if s : [ L ] → [ K ], i : [ K ] → [ L ] and for each x ∈ [ K ] s ( i ( x )) = x and ∀ y < i ( x ) s ( y ) ≤ x . So, i is a left inverse of s and at each x ∈ [ K ] the value i ( x ) is picked only from those elements of s − 1 ( x ) that are “visible from x ,” that is, from those y ′ ∈ s − 1 ( x ) for which s ↾ ( { y : y < y ′ } ) ≤ x . S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 8 / 42
Self-dual Ramsey theorem Definition A pair ( s , i ) is a connection between L and K if s : [ L ] → [ K ], i : [ K ] → [ L ] and for each x ∈ [ K ] s ( i ( x )) = x and ∀ y < i ( x ) s ( y ) ≤ x . So, i is a left inverse of s and at each x ∈ [ K ] the value i ( x ) is picked only from those elements of s − 1 ( x ) that are “visible from x ,” that is, from those y ′ ∈ s − 1 ( x ) for which s ↾ ( { y : y < y ′ } ) ≤ x . We write ( s , i ): [ L ] ↔ [ K ] . S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 8 / 42
Self-dual Ramsey theorem Given connections ( s , i ): [ L ] ↔ [ K ] and ( t , j ): [ M ] ↔ [ L ], define ( s , i ) · ( t , j ): [ M ] ↔ [ K ] as ( s ◦ t , j ◦ i ) . S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 9 / 42
Self-dual Ramsey theorem Given connections ( s , i ): [ L ] ↔ [ K ] and ( t , j ): [ M ] ↔ [ L ], define ( s , i ) · ( t , j ): [ M ] ↔ [ K ] as ( s ◦ t , j ◦ i ) . Theorem (S.) For natural numbers K, L and d > 0 there exists M such that for each d-coloring of all connections between M and K there is ( t 0 , j 0 ): [ M ] ↔ [ L ] such that { ( s , i ) · ( t 0 , j 0 ): ( s , i ): [ L ] ↔ [ K ] } is monochromatic. S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 9 / 42
Algebraic notions Algebraic notions S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 10 / 42
Algebraic notions Abstract Ramsey statement: S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 11 / 42
Algebraic notions Abstract Ramsey statement: given S find F for which F × S ∋ ( f , x ) → f . x ∈ F . S is defined; S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 11 / 42
Algebraic notions Abstract Ramsey statement: given S find F for which F × S ∋ ( f , x ) → f . x ∈ F . S is defined; color F . S ; S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 11 / 42
Algebraic notions Abstract Ramsey statement: given S find F for which F × S ∋ ( f , x ) → f . x ∈ F . S is defined; color F . S ; find f ∈ F with f . S monochromatic. S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 11 / 42
Algebraic notions Abstract Ramsey statement: given S find F for which F × S ∋ ( f , x ) → f . x ∈ F . S is defined; color F . S ; find f ∈ F with f . S monochromatic. Algebraic approach: S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 11 / 42
Algebraic notions Abstract Ramsey statement: given S find F for which F × S ∋ ( f , x ) → f . x ∈ F . S is defined; color F . S ; find f ∈ F with f . S monochromatic. Algebraic approach: multiplication/action S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 11 / 42
Algebraic notions Abstract Ramsey statement: given S find F for which F × S ∋ ( f , x ) → f . x ∈ F . S is defined; color F . S ; find f ∈ F with f . S monochromatic. Algebraic approach: multiplication/action, lifting them to sets S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 11 / 42
Algebraic notions Abstract Ramsey statement: given S find F for which F × S ∋ ( f , x ) → f . x ∈ F . S is defined; color F . S ; find f ∈ F with f . S monochromatic. Algebraic approach: multiplication/action, lifting them to sets, truncation operator. S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 11 / 42
Algebraic notions Multiplicative part S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 12 / 42
Algebraic notions Definition A local actoid S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 13 / 42
Algebraic notions Definition A local actoid consists of two sets A and Z , S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 13 / 42
Algebraic notions Definition A local actoid consists of two sets A and Z , a partial binary function from A × A to A : ( a , b ) → a · b , S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 13 / 42
Algebraic notions Definition A local actoid consists of two sets A and Z , a partial binary function from A × A to A : ( a , b ) → a · b , and a partial binary function from A × Z to Z : ( a , z ) → a . z S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 13 / 42
Algebraic notions Definition A local actoid consists of two sets A and Z , a partial binary function from A × A to A : ( a , b ) → a · b , and a partial binary function from A × Z to Z : ( a , z ) → a . z such that for a , b ∈ A and z ∈ Z if a . ( b . z ) and ( a · b ) . z are both defined, S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 13 / 42
Algebraic notions Definition A local actoid consists of two sets A and Z , a partial binary function from A × A to A : ( a , b ) → a · b , and a partial binary function from A × Z to Z : ( a , z ) → a . z such that for a , b ∈ A and z ∈ Z if a . ( b . z ) and ( a · b ) . z are both defined, then a . ( b . z ) = ( a · b ) . z . S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 13 / 42
Algebraic notions Definition A local actoid ( A , Z ) is called an actoid if for all a , b ∈ A and z ∈ Z , if a . ( b . z ) is defined, then so is ( a · b ) . z . S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 14 / 42
Algebraic notions Definition A local actoid ( A , Z ) is called an actoid if for all a , b ∈ A and z ∈ Z , if a . ( b . z ) is defined, then so is ( a · b ) . z . Note: for a , b , z as above, one has a . ( b . z ) = ( a · b ) . z . S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 14 / 42
Algebraic notions Example. S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 15 / 42
Algebraic notions Example. s : [ L ] → [ K ], t : [ N ] → [ M ] rigid surjections S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 15 / 42
Algebraic notions Example. s : [ L ] → [ K ], t : [ N ] → [ M ] rigid surjections The canonical composition of s and t , denoted by s ◦ t , is defined if L ≤ M . S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 15 / 42
Algebraic notions Example. s : [ L ] → [ K ], t : [ N ] → [ M ] rigid surjections The canonical composition of s and t , denoted by s ◦ t , is defined if L ≤ M . In this case, let s ◦ t be the composition of s with t restricted to the largest initial segment of [ N ] on which this composition is defined. S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 15 / 42
Algebraic notions Example. s : [ L ] → [ K ], t : [ N ] → [ M ] rigid surjections The canonical composition of s and t , denoted by s ◦ t , is defined if L ≤ M . In this case, let s ◦ t be the composition of s with t restricted to the largest initial segment of [ N ] on which this composition is defined. Then s ◦ t : [ N 0 ] → [ K ] is a rigid surjection for some N 0 ≤ N . S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 15 / 42
Algebraic notions A 0 = Z 0 = rigid surjections S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 16 / 42
Algebraic notions A 0 = Z 0 = rigid surjections For s , t ∈ A 0 = Z 0 , let t · s = t . s = s ◦ t whenever s ◦ t is defined. S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 16 / 42
Algebraic notions A 0 = Z 0 = rigid surjections For s , t ∈ A 0 = Z 0 , let t · s = t . s = s ◦ t whenever s ◦ t is defined. ( A 0 , Z 0 ) is a local actoid . S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 16 / 42
Algebraic notions Lifting multiplication and action to sets S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 17 / 42
Algebraic notions Each local actoid ( A , Z ) induces operations on subsets . S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 18 / 42
Algebraic notions Each local actoid ( A , Z ) induces operations on subsets . For F , G ⊆ A , F · G is defined if f · g is defined for all f ∈ F and g ∈ G , and we let F · G = { f · g : f ∈ F , g ∈ G } . S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 18 / 42
Algebraic notions Each local actoid ( A , Z ) induces operations on subsets . For F , G ⊆ A , F · G is defined if f · g is defined for all f ∈ F and g ∈ G , and we let F · G = { f · g : f ∈ F , g ∈ G } . For F ⊆ A and S ⊆ Z , F . S is defined if f . x is defined for all f ∈ F and x ∈ S , and we let F . S = { f . x : f ∈ F , x ∈ S } . S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 18 / 42
Algebraic notions Definition ( A , Z ) a local actoid. S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 19 / 42
Algebraic notions Definition ( A , Z ) a local actoid. Let F be a family of subsets of A and S a family of subsets of Z . S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 19 / 42
Algebraic notions Definition ( A , Z ) a local actoid. Let F be a family of subsets of A and S a family of subsets of Z . We have partial functions from F × F to F and from F × S to S : ( F , G ) → F • G and ( F , S ) → F • S . S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 19 / 42
Algebraic notions Definition ( A , Z ) a local actoid. Let F be a family of subsets of A and S a family of subsets of Z . We have partial functions from F × F to F and from F × S to S : ( F , G ) → F • G and ( F , S ) → F • S . We say that ( F , S ) with these two operations is a local actoid of sets over ( A , Z ) provided that S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 19 / 42
Algebraic notions Definition ( A , Z ) a local actoid. Let F be a family of subsets of A and S a family of subsets of Z . We have partial functions from F × F to F and from F × S to S : ( F , G ) → F • G and ( F , S ) → F • S . We say that ( F , S ) with these two operations is a local actoid of sets over ( A , Z ) provided that whenever F • G is defined, then so is F · G and F • G = F · G , S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 19 / 42
Algebraic notions Definition ( A , Z ) a local actoid. Let F be a family of subsets of A and S a family of subsets of Z . We have partial functions from F × F to F and from F × S to S : ( F , G ) → F • G and ( F , S ) → F • S . We say that ( F , S ) with these two operations is a local actoid of sets over ( A , Z ) provided that whenever F • G is defined, then so is F · G and F • G = F · G , and whenever F • S is defined, then so is F . S and F • S = F . S . S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 19 / 42
Algebraic notions Example. (ctd) S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 20 / 42
Algebraic notions Example. (ctd) ( A 0 , Z 0 ) the local actoid defined earlier S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 20 / 42
Algebraic notions Example. (ctd) ( A 0 , Z 0 ) the local actoid defined earlier F 0 = S 0 consist of sets of the form F L , K = S L , K = { s ∈ A 0 = Z 0 : s : [ L ] → [ K ] } , for L ≥ K > 0. S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 20 / 42
Algebraic notions F N , M • F L , K defined if and only if M = L and F N , L • F L , K = F N , K . S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 21 / 42
Algebraic notions F N , M • F L , K defined if and only if M = L and F N , L • F L , K = F N , K . F N , M • S L , K defined if and only if M = L and F N , L • S L , K = S N , K . S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 21 / 42
Algebraic notions F N , M • F L , K defined if and only if M = L and F N , L • F L , K = F N , K . F N , M • S L , K defined if and only if M = L and F N , L • S L , K = S N , K . ( F 0 , S 0 ) with these operations is an actoid of sets over ( A 0 , Z 0 ). S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 21 / 42
Algebraic notions F N , M • F L , K defined if and only if M = L and F N , L • F L , K = F N , K . F N , M • S L , K defined if and only if M = L and F N , L • S L , K = S N , K . ( F 0 , S 0 ) with these operations is an actoid of sets over ( A 0 , Z 0 ). Note that F N , M · F L , K and F N , M . S L , K are defined if only M ≥ L . S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 21 / 42
Algebraic notions Truncation added S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 22 / 42
Algebraic notions Definition A background is a local actoid ( A , Z ) S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 23 / 42
Algebraic notions Definition A background is a local actoid ( A , Z ) together with a unary function ∂ : Z → Z such that S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 23 / 42
Algebraic notions Definition A background is a local actoid ( A , Z ) together with a unary function ∂ : Z → Z such that for a ∈ A and z ∈ Z , if a . z is defined, then a . ∂ z is defined and a . ∂ z = ∂ ( a . z ) . S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 23 / 42
Algebraic notions Definition A background is a local actoid ( A , Z ) together with a unary function ∂ : Z → Z such that for a ∈ A and z ∈ Z , if a . z is defined, then a . ∂ z is defined and a . ∂ z = ∂ ( a . z ) . ∂ is call a truncation . S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 23 / 42
Algebraic notions Definition A background is a local actoid ( A , Z ) together with a unary function ∂ : Z → Z such that for a ∈ A and z ∈ Z , if a . z is defined, then a . ∂ z is defined and a . ∂ z = ∂ ( a . z ) . ∂ is call a truncation . It is a type of a restriction operator. S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 23 / 42
Algebraic notions Notation: for a background ( A , Z ) with a truncation ∂ and for S ⊆ Z , let ∂ S = { ∂ x : x ∈ S } and, more generally, for t ∈ N ∂ t S = { ∂ t x : x ∈ S } . S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 24 / 42
Algebraic notions Example. (ctd) S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 25 / 42
Algebraic notions Example. (ctd) s : [ L ] → [ K ] a rigid surjection S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 25 / 42
Algebraic notions Example. (ctd) s : [ L ] → [ K ] a rigid surjection If K > 0, then L > 0, and let L 0 = min { y ∈ [ L ]: s ( y ) = K } . S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 25 / 42
Algebraic notions Example. (ctd) s : [ L ] → [ K ] a rigid surjection If K > 0, then L > 0, and let L 0 = min { y ∈ [ L ]: s ( y ) = K } . Define ∂ 0 s = s ↾ [ L 0 − 1] . S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 25 / 42
Algebraic notions Example. (ctd) s : [ L ] → [ K ] a rigid surjection If K > 0, then L > 0, and let L 0 = min { y ∈ [ L ]: s ( y ) = K } . Define ∂ 0 s = s ↾ [ L 0 − 1] . If K = 0, then L = 0 and s is the empty function and we let ∂ 0 ∅ = ∅ . S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 25 / 42
Algebraic notions Example. (ctd) s : [ L ] → [ K ] a rigid surjection If K > 0, then L > 0, and let L 0 = min { y ∈ [ L ]: s ( y ) = K } . Define ∂ 0 s = s ↾ [ L 0 − 1] . If K = 0, then L = 0 and s is the empty function and we let ∂ 0 ∅ = ∅ . ∂ 0 is a truncation forgetting the largest value . S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 25 / 42
Algebraic notions ( A 0 , Z 0 ) the local actoid defined earlier; S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 26 / 42
Algebraic notions ( A 0 , Z 0 ) the local actoid defined earlier; for s ∈ Z 0 , take ∂ 0 s as the truncation. S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 26 / 42
Algebraic notions ( A 0 , Z 0 ) the local actoid defined earlier; for s ∈ Z 0 , take ∂ 0 s as the truncation. ( A 0 , Z 0 ) with ∂ 0 is a background . S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 26 / 42
Abstract pigeonhole and main theorem Abstract pigeonhole and main theorem S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 27 / 42
Abstract pigeonhole and main theorem ( F , S ) a local actoid of sets over a background, S ∈ S S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 28 / 42
Abstract pigeonhole and main theorem ( F , S ) a local actoid of sets over a background, S ∈ S Recall the abstract Ramsey statement : S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 28 / 42
Abstract pigeonhole and main theorem ( F , S ) a local actoid of sets over a background, S ∈ S Recall the abstract Ramsey statement : find F ∈ F for which F • S is defined; color F • S ; find f ∈ F with f . S monochromatic. S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 28 / 42
Abstract pigeonhole and main theorem We consider the equivalence relation ∼ on S given by x 1 ∼ x 2 ⇐ ⇒ ∂ x 1 = ∂ x 2 . S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 29 / 42
Abstract pigeonhole and main theorem We consider the equivalence relation ∼ on S given by x 1 ∼ x 2 ⇐ ⇒ ∂ x 1 = ∂ x 2 . We are looking for a principle of the form : there is F ∈ F such that for each coloring of F • S there is f ∈ F with multiplication by f stabilizing the coloring on equivalence classes of ∼ . S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 29 / 42
Abstract pigeonhole and main theorem Definition Let ( F , S ) be an actoid of sets over a background ( A , Z ). We call ( F , S ) a pigeonhole actoid if S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 30 / 42
Abstract pigeonhole and main theorem Definition Let ( F , S ) be an actoid of sets over a background ( A , Z ). We call ( F , S ) a pigeonhole actoid if (ph) for every d > 0 and S ∈ S there exists F ∈ F such that F • S is defined and for each d -coloring c of F . S there exists f ∈ F such that for all x 1 , x 2 ∈ S we have ∂ x 1 = ∂ x 2 = ⇒ c ( f . x 1 ) = c ( f . x 2 ) . S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 30 / 42
Abstract pigeonhole and main theorem Definition Let ( F , S ) be an actoid of sets over a background ( A , Z ). We call ( F , S ) a pigeonhole actoid if (ph) for every t ≥ 0, d > 0 and S ∈ S there exists F ∈ F such that F • S is defined and for each d -coloring c of F . ∂ t S there exists f ∈ F such that for all x 1 , x 2 ∈ ∂ t S we have ∂ x 1 = ∂ x 2 = ⇒ c ( f . x 1 ) = c ( f . x 2 ) . S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 30 / 42
Abstract pigeonhole and main theorem Definition Let ( F , S ) be an actoid of sets over a background ( A , Z ). We call ( F , S ) a pigeonhole actoid if (ph) for every t ≥ 0, d > 0 and S ∈ S there exists F ∈ F such that F • S is defined and for each d -coloring c of F . ∂ t S there exists f ∈ F such that for all x 1 , x 2 ∈ ∂ t S we have ∂ x 1 = ∂ x 2 = ⇒ c ( f . x 1 ) = c ( f . x 2 ) . (ph): multiplication by f fixes color on equivalence classes of the equivalence relation on ∂ t S given by ∂ x 1 = ∂ x 2 . S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 30 / 42
Abstract pigeonhole and main theorem Definition A family I of subsets of Z for a background ( A , Z ) is called vanishing if for every S ∈ I there is t ∈ N such that ∂ t S consists of at most one element. S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 31 / 42
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