An abstract approach to finite Ramsey theory with applications S - PowerPoint PPT Presentation

Introduction Some consequences of the approach: — new concrete Ramsey results; — a hierarchy of the Ramsey results according to the number of times the abstract Ramsey theorem is applied in their proofs: the classical Ramsey theorem requires one application, the Hales–Jewett theorem requires two, the Graham–Rothschild theorem three, and the new results four; S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 9 / 43

Introduction Some consequences of the approach: — new concrete Ramsey results; — a hierarchy of the Ramsey results according to the number of times the abstract Ramsey theorem is applied in their proofs: the classical Ramsey theorem requires one application, the Hales–Jewett theorem requires two, the Graham–Rothschild theorem three, and the new results four; — a possibility of classifying concrete Ramsey theorems. S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 9 / 43

Introduction The self-dual Ramsey theorem: S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 10 / 43

Introduction The self-dual Ramsey theorem: R a partition of [ n ], C a subset of [ n ], m ∈ N . S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 10 / 43

Introduction The self-dual Ramsey theorem: R a partition of [ n ], C a subset of [ n ], m ∈ N . ( R , C ) is an m - connection if R and C have m elements each and, upon listing R as R 1 , . . . , R m with min R i < min R i +1 and C as c 1 , . . . , c m with c i < c i +1 S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 10 / 43

Introduction The self-dual Ramsey theorem: R a partition of [ n ], C a subset of [ n ], m ∈ N . ( R , C ) is an m - connection if R and C have m elements each and, upon listing R as R 1 , . . . , R m with min R i < min R i +1 and C as c 1 , . . . , c m with c i < c i +1 , we have c i ∈ R i for i ≤ m and c i < min R i +1 for i < m . S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 10 / 43

Introduction The self-dual Ramsey theorem: R a partition of [ n ], C a subset of [ n ], m ∈ N . ( R , C ) is an m - connection if R and C have m elements each and, upon listing R as R 1 , . . . , R m with min R i < min R i +1 and C as c 1 , . . . , c m with c i < c i +1 , we have c i ∈ R i for i ≤ m and c i < min R i +1 for i < m . An l -connection ( Q , B ) is an l - subconnection of an m -connection ( R , C ) S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 10 / 43

Introduction The self-dual Ramsey theorem: R a partition of [ n ], C a subset of [ n ], m ∈ N . ( R , C ) is an m - connection if R and C have m elements each and, upon listing R as R 1 , . . . , R m with min R i < min R i +1 and C as c 1 , . . . , c m with c i < c i +1 , we have c i ∈ R i for i ≤ m and c i < min R i +1 for i < m . An l -connection ( Q , B ) is an l - subconnection of an m -connection ( R , C ) if R is a coarser partition than Q and B ⊆ C . S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 10 / 43

Introduction Theorem (S.) Let d > 0 . For k , l ∈ N there exists m ∈ N such that for each d-coloring of all k-subconnections of an m-connection ( R , C ) S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 11 / 43

Introduction Theorem (S.) Let d > 0 . For k , l ∈ N there exists m ∈ N such that for each d-coloring of all k-subconnections of an m-connection ( R , C ) there exists an l-subconnection ( Q , B ) of ( R , C ) such that all k-subconnections of ( Q , B ) get the same color. S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 11 / 43

Algebraic notions Algebraic notions S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 12 / 43

Algebraic notions Abstract Ramsey statement: S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 13 / 43

Algebraic notions Abstract Ramsey statement: given P find F for which F × P ∋ ( f , x ) → f . x ∈ F . P is defined; S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 13 / 43

Algebraic notions Abstract Ramsey statement: given P find F for which F × P ∋ ( f , x ) → f . x ∈ F . P is defined; color F . P ; S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 13 / 43

Algebraic notions Abstract Ramsey statement: given P find F for which F × P ∋ ( f , x ) → f . x ∈ F . P is defined; color F . P ; find f 0 ∈ F with f 0 . P monochromatic. S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 13 / 43

Algebraic notions Restatement of the classical Ramsey theorem: S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 14 / 43

Algebraic notions Restatement of the classical Ramsey theorem: Identify p element subsets of [ q ] with increasing injections from [ p ] to [ q ]. S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 14 / 43

Algebraic notions Restatement of the classical Ramsey theorem: Identify p element subsets of [ q ] with increasing injections from [ p ] to [ q ]. Fix natural numbers d and k ≤ l . S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 14 / 43

Algebraic notions Restatement of the classical Ramsey theorem: Identify p element subsets of [ q ] with increasing injections from [ p ] to [ q ]. Fix natural numbers d and k ≤ l . Let P be the set of all increasing injections from [ k ] to [ l ]. S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 14 / 43

Algebraic notions Restatement of the classical Ramsey theorem: Identify p element subsets of [ q ] with increasing injections from [ p ] to [ q ]. Fix natural numbers d and k ≤ l . Let P be the set of all increasing injections from [ k ] to [ l ]. There is an m such that S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 14 / 43

Algebraic notions Restatement of the classical Ramsey theorem: Identify p element subsets of [ q ] with increasing injections from [ p ] to [ q ]. Fix natural numbers d and k ≤ l . Let P be the set of all increasing injections from [ k ] to [ l ]. There is an m such that for the set F of all increasing injections from [ l ] to [ m ], if we d -color the set { f ◦ x : f ∈ F , x ∈ P } = all increasing injections from [ k ] to [ m ] , S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 14 / 43

Algebraic notions Restatement of the classical Ramsey theorem: Identify p element subsets of [ q ] with increasing injections from [ p ] to [ q ]. Fix natural numbers d and k ≤ l . Let P be the set of all increasing injections from [ k ] to [ l ]. There is an m such that for the set F of all increasing injections from [ l ] to [ m ], if we d -color the set { f ◦ x : f ∈ F , x ∈ P } = all increasing injections from [ k ] to [ m ] , then there exists f 0 ∈ F such that { f 0 ◦ x : x ∈ P } is monochromatic. S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 14 / 43

Algebraic notions Normed backgrounds S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 15 / 43

Algebraic notions Let ( A , · , ∂, | · | , o ) be such that S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 16 / 43

Algebraic notions Let ( A , · , ∂, | · | , o ) be such that — · is a partial function from A × A to A ( multiplication ); S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 16 / 43

Algebraic notions Let ( A , · , ∂, | · | , o ) be such that — · is a partial function from A × A to A ( multiplication ); — ∂ is a function from A to A ( truncation ); S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 16 / 43

Algebraic notions Let ( A , · , ∂, | · | , o ) be such that — · is a partial function from A × A to A ( multiplication ); — ∂ is a function from A to A ( truncation ); — | · | is a function from A to a linearly ordered set ( L , ≤ ) ( norm ); S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 16 / 43

Algebraic notions Let ( A , · , ∂, | · | , o ) be such that — · is a partial function from A × A to A ( multiplication ); — ∂ is a function from A to A ( truncation ); — | · | is a function from A to a linearly ordered set ( L , ≤ ) ( norm ); — o is an element of A ( zero ). S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 16 / 43

Algebraic notions Such a structure with associative multiplication is called a normed background provided that for a , b , c ∈ A : S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 17 / 43

Algebraic notions Such a structure with associative multiplication is called a normed background provided that for a , b , c ∈ A : (i) if a · b and a · ∂ b are defined, then ∂ ( a · b ) = a · ∂ b ; S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 17 / 43

Algebraic notions Such a structure with associative multiplication is called a normed background provided that for a , b , c ∈ A : (i) if a · b and a · ∂ b are defined, then ∂ ( a · b ) = a · ∂ b ; (ii) | ∂ a | ≤ | a | ; S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 17 / 43

Algebraic notions Such a structure with associative multiplication is called a normed background provided that for a , b , c ∈ A : (i) if a · b and a · ∂ b are defined, then ∂ ( a · b ) = a · ∂ b ; (ii) | ∂ a | ≤ | a | ; (iii) if | b | ≤ | c | and a · c is defined, then so is a · b and | a · b | ≤ | a · c | ; S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 17 / 43

Algebraic notions Such a structure with associative multiplication is called a normed background provided that for a , b , c ∈ A : (i) if a · b and a · ∂ b are defined, then ∂ ( a · b ) = a · ∂ b ; (ii) | ∂ a | ≤ | a | ; (iii) if | b | ≤ | c | and a · c is defined, then so is a · b and | a · b | ≤ | a · c | ; (iv) there is t = t a ∈ N with ∂ t a = o , and ∂ o = o . S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 17 / 43

Algebraic notions Example. S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 18 / 43

Algebraic notions Example. A = the set of all strictly increasing functions from [ k ] = { 1 , . . . , k } to N \ { 0 } . S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 18 / 43

Algebraic notions Example. A = the set of all strictly increasing functions from [ k ] = { 1 , . . . , k } to N \ { 0 } . For a , b ∈ A with a : [ k ] → N \ { 0 } and b : [ l ] → N \ { 0 } , S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 18 / 43

Algebraic notions Example. A = the set of all strictly increasing functions from [ k ] = { 1 , . . . , k } to N \ { 0 } . For a , b ∈ A with a : [ k ] → N \ { 0 } and b : [ l ] → N \ { 0 } , a · b defined when [ k ] contains the image of b and a · b = a ◦ b . S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 18 / 43

Algebraic notions For a ∈ A with a : [ k ] → N \ { 0 } , let S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 19 / 43

Algebraic notions For a ∈ A with a : [ k ] → N \ { 0 } , let ∂ a = a ↾ [ k − 1] . S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 19 / 43

Algebraic notions For a ∈ A with a : [ k ] → N \ { 0 } , let ∂ a = a ↾ [ k − 1] . Define | · | : A → N , where N is taken with its natural linear order. For a ∈ A with a : [ k ] → N \ { 0 } , let S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 19 / 43

Algebraic notions For a ∈ A with a : [ k ] → N \ { 0 } , let ∂ a = a ↾ [ k − 1] . Define | · | : A → N , where N is taken with its natural linear order. For a ∈ A with a : [ k ] → N \ { 0 } , let � a ( k ) , if k > 0; | a | = 0 , if k = 0. S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 19 / 43

Algebraic notions For a ∈ A with a : [ k ] → N \ { 0 } , let ∂ a = a ↾ [ k − 1] . Define | · | : A → N , where N is taken with its natural linear order. For a ∈ A with a : [ k ] → N \ { 0 } , let � a ( k ) , if k > 0; | a | = 0 , if k = 0. Let o ∈ A be the empty function. S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 19 / 43

Algebraic notions For a ∈ A with a : [ k ] → N \ { 0 } , let ∂ a = a ↾ [ k − 1] . Define | · | : A → N , where N is taken with its natural linear order. For a ∈ A with a : [ k ] → N \ { 0 } , let � a ( k ) , if k > 0; | a | = 0 , if k = 0. Let o ∈ A be the empty function. A with the above defined operations is a normed background. S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 19 / 43

Algebraic notions Lifting multiplication to sets S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 20 / 43

Algebraic notions Each normed background A induces multiplication on subsets . S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 21 / 43

Algebraic notions Each normed background A induces multiplication 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 January 2013 21 / 43

Algebraic notions Definition A a normed background. Let F be a family of subsets of A . S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 22 / 43

Algebraic notions Definition A a normed background. Let F be a family of subsets of A . Assume we have a partial function from F × F to F : ( F , G ) → F • G . S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 22 / 43

Algebraic notions Definition A a normed background. Let F be a family of subsets of A . Assume we have a partial function from F × F to F : ( F , G ) → F • G . We say that F with this operation is a family over A provided that S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 22 / 43

Algebraic notions Definition A a normed background. Let F be a family of subsets of A . Assume we have a partial function from F × F to F : ( F , G ) → F • G . We say that F with this operation is a family over A 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 January 2013 22 / 43

Algebraic notions Example. (ctd) S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 23 / 43

Algebraic notions Example. (ctd) For k , l ∈ N with k ≤ l , let � l � = the set of all increasing functions from [ k ] to [ l ] . k � l � can be identified with the set of all k element subsets of [ l ]. k S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 23 / 43

Algebraic notions Example. (ctd) For k , l ∈ N with k ≤ l , let � l � = the set of all increasing functions from [ k ] to [ l ] . k � l � can be identified with the set of all k element subsets of [ l ]. k Let � l � F = all with k ≤ l . k S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 23 / 43

Algebraic notions Example. (ctd) For k , l ∈ N with k ≤ l , let � l � = the set of all increasing functions from [ k ] to [ l ] . k � l � can be identified with the set of all k element subsets of [ l ]. k Let � l � F = all with k ≤ l . k � n � l � � • on F to be defined when m = l and Declare m k � n � � l � � n � • = . l k k S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 23 / 43

Algebraic notions Example. (ctd) For k , l ∈ N with k ≤ l , let � l � = the set of all increasing functions from [ k ] to [ l ] . k � l � can be identified with the set of all k element subsets of [ l ]. k Let � l � F = all with k ≤ l . k � n � l � � • on F to be defined when m = l and Declare m k � n � � l � � n � • = . l k k � n � l � n � l � � � � Clear: • = · . l k l k S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 23 / 43

Abstract Ramsey and abstract pigeonhole statements Abstract Ramsey and abstract pigeonhole statements S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 24 / 43

Abstract Ramsey and abstract pigeonhole statements Ramsey statement S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 25 / 43

Abstract Ramsey and abstract pigeonhole statements F a family over a normed background S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 26 / 43

Abstract Ramsey and abstract pigeonhole statements F a family over a normed background The following condition is our Ramsey statement: S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 26 / 43

Abstract Ramsey and abstract pigeonhole statements F a family over a normed background The following condition is our Ramsey statement: (R) given d > 0, for each P ∈ F , S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 26 / 43

Abstract Ramsey and abstract pigeonhole statements F a family over a normed background The following condition is our Ramsey statement: (R) given d > 0, for each P ∈ F , there is F ∈ F such that S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 26 / 43

Abstract Ramsey and abstract pigeonhole statements F a family over a normed background The following condition is our Ramsey statement: (R) given d > 0, for each P ∈ F , there is F ∈ F such that F • P is defined and S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 26 / 43

Abstract Ramsey and abstract pigeonhole statements F a family over a normed background The following condition is our Ramsey statement: (R) given d > 0, for each P ∈ F , there is F ∈ F such that F • P is defined and for every d -coloring of F • P S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 26 / 43

Abstract Ramsey and abstract pigeonhole statements F a family over a normed background The following condition is our Ramsey statement: (R) given d > 0, for each P ∈ F , there is F ∈ F such that F • P is defined and for every d -coloring of F • P there is f ∈ F such that f · P is monochromatic. S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 26 / 43

Abstract Ramsey and abstract pigeonhole statements Example. (ctd) S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 27 / 43

Abstract Ramsey and abstract pigeonhole statements Example. (ctd) Condition (R): S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 27 / 43

Abstract Ramsey and abstract pigeonhole statements Example. (ctd) Condition (R): given d > 0 and k ≤ l S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 27 / 43

Abstract Ramsey and abstract pigeonhole statements Example. (ctd) Condition (R): given d > 0 and k ≤ l there exists m ≥ l S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 27 / 43

Abstract Ramsey and abstract pigeonhole statements Example. (ctd) Condition (R): given d > 0 and k ≤ l there exists m ≥ l such that � m � l � m � � � for each d -coloring of • = l k k S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 27 / 43

Abstract Ramsey and abstract pigeonhole statements Example. (ctd) Condition (R): given d > 0 and k ≤ l there exists m ≥ l such that � m � l � m � � � for each d -coloring of • = l k k � m � there exists a ∈ l S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 27 / 43

Abstract Ramsey and abstract pigeonhole statements Example. (ctd) Condition (R): given d > 0 and k ≤ l there exists m ≥ l such that � m � l � m � � � for each d -coloring of • = l k k � m � there exists a ∈ such that l � l � { a ◦ x : x ∈ } is monochromatic. k S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 27 / 43

Abstract Ramsey and abstract pigeonhole statements Example. (ctd) Condition (R): given d > 0 and k ≤ l there exists m ≥ l such that � m � l � m � � � for each d -coloring of • = l k k � m � there exists a ∈ such that l � l � { a ◦ x : x ∈ } is monochromatic. k This is the classical Ramsey theorem. S� lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 27 / 43