ramsey classes by partite construction i
play

Ramsey Classes by Partite Construction I Honza Hubi cka - PowerPoint PPT Presentation

Jul 04, 2023 •293 likes •1.49k views

Ramsey Classes by Partite Construction I Honza Hubi cka Mathematics and Statistics University of Calgary Calgary Institute of Computer Science Charles University Prague Joint work with Jaroslav Neet ril Permutation Groups and

  1. The Partite Construction A = C =? B = Construction outline: Put n such → ( | B | ) | A | n − 2 . (For every coloring of | A | tuples in { 1 , 2 , . . . n } there exists monochromatic subset of size | B | ). Here n = 6. Picture 0: | K | n -partite system P 0 s.t. for every coloring of copies of A in P 0 where the color of a copy � A depends only on a projection π ( � A ) there exists a monochromatic copy of B . Enumerate by A 1 , . . . A N all possible projections of copies of A in P 0 . Pictures 1. . . n: K n -partite systems P 1 , . . . P N s.t. for every coloring of copies of A in P i there exists a copy of P i − 1 where all copies of A with projection A i are monochromatic. J. Hubiˇ cka Ramsey Classes by Partite Construction I

  2. The Partite Construction: Picture 0 Picture 0: K n -partite system P 0 s.t. for every coloring of copies of A in P 0 where the color of a copy � A depends only on a projection π ( � A ) there exists a monochromatic copy of B . A = B = J. Hubiˇ cka Ramsey Classes by Partite Construction I

  3. The Partite Construction: Picture 0 Picture 0: K n -partite system P 0 s.t. for every coloring of copies of A in P 0 where the color of a copy � A depends only on a projection π ( � A ) there exists a monochromatic copy of B . A = B = J. Hubiˇ cka Ramsey Classes by Partite Construction I

  4. The Partite Construction: Picture 0 Picture 0: K n -partite system P 0 s.t. for every coloring of copies of A in P 0 where the color of a copy � A depends only on a projection π ( � A ) there exists a monochromatic copy of B . A = B = J. Hubiˇ cka Ramsey Classes by Partite Construction I

  5. The Partite Construction: Picture 1 Picture 1: K n -partite system P 1 s.t. for every coloring of copies of A in P 1 there exists a copy of P 0 where all copies of A with projection A 1 are monochromatic. A = B = J. Hubiˇ cka Ramsey Classes by Partite Construction I

  6. The Partite Construction: Picture 1 Picture 1: K n -partite system P 1 s.t. for every coloring of copies of A in P 1 there exists a copy of P 0 where all copies of A with projection A 1 are monochromatic. A = B = J. Hubiˇ cka Ramsey Classes by Partite Construction I

  7. The Partite Construction: Picture 1 Picture 1: K n -partite system P 1 s.t. for every coloring of copies of A in P 1 there exists a copy of P 0 where all copies of A with projection A 1 are monochromatic. A = B = J. Hubiˇ cka Ramsey Classes by Partite Construction I

  8. The Partite Construction: Picture 1 Picture 1: K n -partite system P 1 s.t. for every coloring of copies of A in P 1 there exists a copy of P 0 where all copies of A with projection A 1 are monochromatic. A = B = J. Hubiˇ cka Ramsey Classes by Partite Construction I

  9. The Partite Construction: Picture 1 Picture 1: K n -partite system P 1 s.t. for every coloring of copies of A in P 1 there exists a copy of P 0 where all copies of A with projection A 1 are monochromatic. A = B = J. Hubiˇ cka Ramsey Classes by Partite Construction I

  10. The Partite Construction: Picture 1 Picture 1: K n -partite system P 1 s.t. for every coloring of copies of A in P 1 there exists a copy of P 0 where all copies of A with projection A 1 are monochromatic. A = B = J. Hubiˇ cka Ramsey Classes by Partite Construction I

  11. The Partite Construction: Picture 1 Picture 1: K n -partite system P 1 s.t. for every coloring of copies of A in P 1 there exists a copy of P 0 where all copies of A with projection A 1 are monochromatic. A = B = J. Hubiˇ cka Ramsey Classes by Partite Construction I

  12. The Partite Construction: Picture 1 Picture 1: K n -partite system P 1 s.t. for every coloring of copies of A in P 1 there exists a copy of P 0 where all copies of A with projection A 1 are monochromatic. A = B = J. Hubiˇ cka Ramsey Classes by Partite Construction I

  13. The Partite Construction: Picture 2 Picture 2: K n -partite system P 2 s.t. for every coloring of copies of A in P 2 there exists a copy of P 1 where all copies of A with projection A 2 are monochromatic. J. Hubiˇ cka Ramsey Classes by Partite Construction I

  14. The Partite Construction: Picture 2 Picture 2: K n -partite system P 2 s.t. for every coloring of copies of A in P 2 there exists a copy of P 1 where all copies of A with projection A 2 are monochromatic. A = B = J. Hubiˇ cka Ramsey Classes by Partite Construction I

  15. The Partite Construction: Picture 2 Picture 2: K n -partite system P 2 s.t. for every coloring of copies of A in P 2 there exists a copy of P 1 where all copies of A with projection A 2 are monochromatic. A = B = J. Hubiˇ cka Ramsey Classes by Partite Construction I

  16. The Partite Construction: Picture 2 Picture 2: K n -partite system P 2 s.t. for every coloring of copies of A in P 2 there exists a copy of P 1 where all copies of A with projection A 2 are monochromatic. A = B = J. Hubiˇ cka Ramsey Classes by Partite Construction I

  17. The Partite Construction: Picture 2 Picture 2: K n -partite system P 2 s.t. for every coloring of copies of A in P 2 there exists a copy of P 1 where all copies of A with projection A 2 are monochromatic. A = B = J. Hubiˇ cka Ramsey Classes by Partite Construction I

  18. The Partite Construction: Picture 2 Picture 2: K n -partite system P 2 s.t. for every coloring of copies of A in P 2 there exists a copy of P 1 where all copies of A with projection A 2 are monochromatic. A = B = J. Hubiˇ cka Ramsey Classes by Partite Construction I

  19. The Partite Construction: Summary Ramsey Theorem: K | A | K n − → ( K | B | ) 2 J. Hubiˇ cka Ramsey Classes by Partite Construction I

  20. The Partite Construction: Summary Ramsey Theorem: K | A | K n − → ( K | B | ) 2 Construct P 0 J. Hubiˇ cka Ramsey Classes by Partite Construction I

  21. The Partite Construction: Summary Ramsey Theorem: K | A | K n − → ( K | B | ) 2 Construct P 0 Enumerate by A 1 , . . . , A N all possible projections of copies of A in P 0 Construct P 1 , . . . , P N J. Hubiˇ cka Ramsey Classes by Partite Construction I

  22. The Partite Construction: Summary Ramsey Theorem: K | A | K n − → ( K | B | ) 2 Construct P 0 Enumerate by A 1 , . . . , A N all possible projections of copies of A in P 0 Construct P 1 , . . . , P N B i : partite system induced on P i − 1 by all copies of all with projection to A i → ( B i ) A i Partite lemma: C i − 2 P i is built repeated free amalgamation of P i over all copies of B i in C i J. Hubiˇ cka Ramsey Classes by Partite Construction I

  23. The Partite Construction: Summary Ramsey Theorem: K | A | K n − → ( K | B | ) 2 Construct P 0 Enumerate by A 1 , . . . , A N all possible projections of copies of A in P 0 Construct P 1 , . . . , P N B i : partite system induced on P i − 1 by all copies of all with projection to A i → ( B i ) A i Partite lemma: C i − 2 P i is built repeated free amalgamation of P i over all copies of B i in C i Put C = P N J. Hubiˇ cka Ramsey Classes by Partite Construction I

  24. The Partite Lemma Lemma Let A be a structure s.t. A = { 1 , 2 , . . . , a } and B be an A -partite system. Then there exists a A -partite system C s.t. → ( B ) A C − 2 . J. Hubiˇ cka Ramsey Classes by Partite Construction I

  25. The Partite Lemma Lemma Let A be a structure s.t. A = { 1 , 2 , . . . , a } and B be an A -partite system. Then there exists a A -partite system C s.t. → ( B ) A C − 2 . X 1 1 a b B A = B = X 2 x y z 2 B J. Hubiˇ cka Ramsey Classes by Partite Construction I

  26. The Partite Lemma Proof by application of Hales-Jewett theorem Theorem (Hales-Jewett theorem) For every finite alphabet Σ there exists N = HJ (Σ) so that for every 2 -coloring of functions h : { 1 , 2 , . . . , N } → Σ there exists a monochromatic combinatorial line. J. Hubiˇ cka Ramsey Classes by Partite Construction I

  27. The Partite Lemma Proof by application of Hales-Jewett theorem Theorem (Hales-Jewett theorem) For every finite alphabet Σ there exists N = HJ (Σ) so that for every 2 -coloring of functions h : { 1 , 2 , . . . , N } → Σ there exists a monochromatic combinatorial line. Definition For non-empty ω ⊆ { 1 , 2 , . . . , N } and f : { 1 , 2 , . . . , N } \ ω → Σ combinatorial line ( ω, f ) is the set of all functions f ′ : { 1 , 2 , . . . , N } → Σ such that � constant for i ∈ ω , f ′ ( i ) = f ( i ) otherwise. J. Hubiˇ cka Ramsey Classes by Partite Construction I

  28. The Partite Lemma Proof by application of Hales-Jewett theorem X 1 a 1 b B A = B = X 2 x y z 2 B J. Hubiˇ cka Ramsey Classes by Partite Construction I

  29. The Partite Lemma Proof by application of Hales-Jewett theorem X 1 a 1 b B A = B = X 2 x y z 2 B � a � � b � � b � Σ = { } (alphabet describe all copies of A in B ) , , x y z J. Hubiˇ cka Ramsey Classes by Partite Construction I

  30. The Partite Lemma Proof by application of Hales-Jewett theorem X 1 a 1 b B A = B = X 2 x y z 2 B � a � � b � � b � Σ = { } (alphabet describe all copies of A in B ) , , x y z N = HJ (Σ) J. Hubiˇ cka Ramsey Classes by Partite Construction I

  31. The Partite Lemma Proof by application of Hales-Jewett theorem X 1 a 1 b B A = B = X 2 x y z 2 B � a � � b � � b � Σ = { } (alphabet describe all copies of A in B ) , , x y z N = HJ (Σ) Build C so that functions h : { 1 , 2 , . . . , N } → Σ correspond to copies of A and combinatorial lines to copies of B : J. Hubiˇ cka Ramsey Classes by Partite Construction I

  32. The Partite Lemma Proof by application of Hales-Jewett theorem X 1 a 1 b B A = B = X 2 x y z 2 B � a � � b � � b � Σ = { } (alphabet describe all copies of A in B ) , , x y z N = HJ (Σ) Build C so that functions h : { 1 , 2 , . . . , N } → Σ correspond to copies of A and combinatorial lines to copies of B : Vertices in partition X i C : Functions f : { 1 , 2 , . . . , N } → X i B . J. Hubiˇ cka Ramsey Classes by Partite Construction I

  33. The Partite Lemma Proof by application of Hales-Jewett theorem X 1 a 1 b B A = B = X 2 x y z 2 B � a � � b � � b � Σ = { } (alphabet describe all copies of A in B ) , , x y z N = HJ (Σ) Build C so that functions h : { 1 , 2 , . . . , N } → Σ correspond to copies of A and combinatorial lines to copies of B : Vertices in partition X i C : Functions f : { 1 , 2 , . . . , N } → X i B . Intended embedding B → C corresponding the combinatorial line ( ω, f ) : � v for i ∈ ω , e ω, f ( v )( i ) vertex of f ( i ) in the same partition as v otherwise. J. Hubiˇ cka Ramsey Classes by Partite Construction I

  34. The Partite Lemma Proof by application of Hales-Jewett theorem X 1 a 1 b B A = B = X 2 x y z 2 B � a � � b � � b � Σ = { } (alphabet describe all copies of A in B ) , , x y z N = HJ (Σ) Build C so that functions h : { 1 , 2 , . . . , N } → Σ correspond to copies of A and combinatorial lines to copies of B : Vertices in partition X i C : Functions f : { 1 , 2 , . . . , N } → X i B . Intended embedding B → C corresponding the combinatorial line ( ω, f ) : � v for i ∈ ω , e ω, f ( v )( i ) vertex of f ( i ) in the same partition as v otherwise. Fact: It is possible to add tuples to relations as needed to make this work J. Hubiˇ cka Ramsey Classes by Partite Construction I

  35. The Partite Lemma Easy description of C : X 1 1 a b B A = B = X 2 x y z 2 B J. Hubiˇ cka Ramsey Classes by Partite Construction I

  36. The Partite Lemma Easy description of C : X 1 1 a b B A = B = X 2 x y z 2 B Vertices in partition X i C : Functions f : { 1 , 2 , . . . , N } → X i B Add as many tuples to relations as possible such that all the evaluation maps g i ( f ) = f ( i ) are homomorphisms from C to B . J. Hubiˇ cka Ramsey Classes by Partite Construction I

  37. Uses of the partite construction Let class K be a class of structures satisfying given axioms. To show that K is Ramsey one can show that the partite construction preserve the axioms. J. Hubiˇ cka Ramsey Classes by Partite Construction I

  38. Uses of the partite construction Let class K be a class of structures satisfying given axioms. To show that K is Ramsey one can show that the partite construction preserve the axioms. Nešetˇ ril, Rödl, 1977: Classes with forbidden 1 (amalgamation) irreducible structures J. Hubiˇ cka Ramsey Classes by Partite Construction I

  39. Uses of the partite construction Let class K be a class of structures satisfying given axioms. To show that K is Ramsey one can show that the partite construction preserve the axioms. Nešetˇ ril, Rödl, 1977: Classes with forbidden 1 (amalgamation) irreducible structures Nešetˇ ril, Rödl, 1984: Acyclic graphs and partial orders with 2 linear extension J. Hubiˇ cka Ramsey Classes by Partite Construction I

  40. Uses of the partite construction Let class K be a class of structures satisfying given axioms. To show that K is Ramsey one can show that the partite construction preserve the axioms. Nešetˇ ril, Rödl, 1977: Classes with forbidden 1 (amalgamation) irreducible structures Nešetˇ ril, Rödl, 1984: Acyclic graphs and partial orders with 2 linear extension Nešetˇ ril, 2005: Metric spaces 3 J. Hubiˇ cka Ramsey Classes by Partite Construction I

  41. Uses of the partite construction Let class K be a class of structures satisfying given axioms. To show that K is Ramsey one can show that the partite construction preserve the axioms. Nešetˇ ril, Rödl, 1977: Classes with forbidden 1 (amalgamation) irreducible structures Nešetˇ ril, Rödl, 1984: Acyclic graphs and partial orders with 2 linear extension Nešetˇ ril, 2005: Metric spaces 3 Nešetˇ ril, 2010–: Classes with finitely many forbidden 4 homomorphisms J. Hubiˇ cka Ramsey Classes by Partite Construction I

  42. Uses of the partite construction Let class K be a class of structures satisfying given axioms. To show that K is Ramsey one can show that the partite construction preserve the axioms. Nešetˇ ril, Rödl, 1977: Classes with forbidden 1 (amalgamation) irreducible structures Nešetˇ ril, Rödl, 1984: Acyclic graphs and partial orders with 2 linear extension Nešetˇ ril, 2005: Metric spaces 3 Nešetˇ ril, 2010–: Classes with finitely many forbidden 4 homomorphisms H., Nešetˇ ril, 2014: Classes with unary algebraic closure 5 J. Hubiˇ cka Ramsey Classes by Partite Construction I

  43. Uses of the partite construction Let class K be a class of structures satisfying given axioms. To show that K is Ramsey one can show that the partite construction preserve the axioms. Nešetˇ ril, Rödl, 1977: Classes with forbidden 1 (amalgamation) irreducible structures Nešetˇ ril, Rödl, 1984: Acyclic graphs and partial orders with 2 linear extension Nešetˇ ril, 2005: Metric spaces 3 Nešetˇ ril, 2010–: Classes with finitely many forbidden 4 homomorphisms H., Nešetˇ ril, 2014: Classes with unary algebraic closure 5 H., Nešetˇ ril, 2015–: (some) classes non-unary algebraic 6 closure J. Hubiˇ cka Ramsey Classes by Partite Construction I

  44. Uses of the partite construction Let class K be a class of structures satisfying given axioms. To show that K is Ramsey one can show that the partite construction preserve the axioms. Nešetˇ ril, Rödl, 1977: Classes with forbidden 1 (amalgamation) irreducible structures Nešetˇ ril, Rödl, 1984: Acyclic graphs and partial orders with 2 linear extension Nešetˇ ril, 2005: Metric spaces 3 Nešetˇ ril, 2010–: Classes with finitely many forbidden 4 homomorphisms H., Nešetˇ ril, 2014: Classes with unary algebraic closure 5 H., Nešetˇ ril, 2015–: (some) classes non-unary algebraic 6 closure H., Nešetˇ ril, 2014–: Classes with infinitely many forbidden 7 homomorphisms. J. Hubiˇ cka Ramsey Classes by Partite Construction I

  45. The Partite Construction Ramsey Theorem: K | A | K n − → ( K | B | ) 2 Construct P 0 Enumerate by A 1 , . . . , A N all possible projections of copies of A in P 0 Construct P 1 , . . . , P N B i : partite system induced on P i − 1 by all copies of all with projection to A i → ( B i ) A i Partite lemma: C i − 2 P i is built repeated free amalgamation of P i over all copies of B i in C i Put C = P N J. Hubiˇ cka Ramsey Classes by Partite Construction I

  46. The Induced Partite Construction Nešetˇ ril-Rödl Theorem: → ( B ) A C 0 − 2 Construct C 0 -partite P 0 Enumerate by A 1 , . . . , A N all possible projections of copies of A in P 0 Construct C 0 -partite P 1 , . . . , P N : B i : partite system induced on P i − 1 by all copies of all with projection to A i → ( B i ) A i Partite lemma: C i − 2 P i is built repeated free amalgamation of P i over all copies of B i in C i Put C = P N J. Hubiˇ cka Ramsey Classes by Partite Construction I

  47. The Induced Partite Construction Nešetˇ ril-Rödl Theorem: → ( B ) A C 0 − 2 Construct C 0 -partite P 0 Enumerate by A 1 , . . . , A N all possible projections of copies of A in P 0 Construct C 0 -partite P 1 , . . . , P N : B i : partite system induced on P i − 1 by all copies of all with projection to A i → ( B i ) A i Partite lemma: C i − 2 P i is built repeated free amalgamation of P i over all copies of B i in C i Put C = P N If K is irreducible and A , B are K -free, then so is C . J. Hubiˇ cka Ramsey Classes by Partite Construction I

  48. An exotic example Bow-tie graph: J. Hubiˇ cka Ramsey Classes by Partite Construction I

  49. An exotic example Bow-tie graph: Amalgamation of two triangles must unify vertices. Wrong! J. Hubiˇ cka Ramsey Classes by Partite Construction I

  50. Structure of bow-tie-free graphs Structure of bow-tie-free graphs Edges in no triangles Edges in 1 triangle Edges in 2 + triangles J. Hubiˇ cka Ramsey Classes by Partite Construction I

  51. Structure of bow-tie-free graphs Structure of bow-tie-free graphs Edges in no triangles Edges in 1 triangle Edges in 2 + triangles Definition Chimney is a graph created by gluing multiple triangles over one edge. Definition Graph is good if every vertex is either in a chimney or K 4 . J. Hubiˇ cka Ramsey Classes by Partite Construction I

  52. Structure of bow-tie-free graphs J. Hubiˇ cka Ramsey Classes by Partite Construction I

  53. Structure of bow-tie-free graphs Graph is good if every vertex is either in a chimney or K 4 . J. Hubiˇ cka Ramsey Classes by Partite Construction I

  54. Structure of bow-tie-free graphs Graph is good if every vertex is either in a chimney or K 4 . Every bowtie-free graph G is a subgraph of some good graph G ′ . J. Hubiˇ cka Ramsey Classes by Partite Construction I

  55. Structure of bow-tie-free graphs Graph is good if every vertex is either in a chimney or K 4 . Every bowtie-free graph G is a subgraph of some good graph G ′ . For every good graph G = ( V , E ) the graph G ∆ = ( V , E ∆ ) ( E ∆ are edges in triangles) is a disjoint union of copies of chimneys and K 4 . J. Hubiˇ cka Ramsey Classes by Partite Construction I

  56. Structure of bow-tie-free graphs Graph is good if every vertex is either in a chimney or K 4 . Every bowtie-free graph G is a subgraph of some good graph G ′ . For every good graph G = ( V , E ) the graph G ∆ = ( V , E ∆ ) ( E ∆ are edges in triangles) is a disjoint union of copies of chimneys and K 4 . Closure of a vertex v = all endpoints of red edges contained in triangles containing v . Lemma Bow-tie-free graphs have free amalgamation over closed structures. J. Hubiˇ cka Ramsey Classes by Partite Construction I

  57. Ramsey property of bow-tie free graphs 3 types of vertices and their closures: J. Hubiˇ cka Ramsey Classes by Partite Construction I

  58. Ramsey property of bow-tie free graphs 3 types of vertices and their closures: To describe lift of bowtie graphs we only need to forbid all triangles except for B-B-R and R-R-R. J. Hubiˇ cka Ramsey Classes by Partite Construction I

  59. Ramsey property of bow-tie free graphs 3 types of vertices and their closures: To describe lift of bowtie graphs we only need to forbid all triangles except for B-B-R and R-R-R. Theorem (H., Nešetˇ ril, 2014) The class of graphs not containing bow-tie as non-induced subgraph have Ramsey lift. J. Hubiˇ cka Ramsey Classes by Partite Construction I

  60. Unary closures = relations with out-degree 1 Unary closure description C is a set of pairs ( R U , R B ) where R U is unary relation and R B is binary relation. We say that structure A is C -closed if for every pair ( R U , R B ) the B -outdegree of every vertex of A that is in U is 1. Theorem (H., Nešetˇ ril, 2015) Let E be a family of complete ordered structures and U an unary closure description. Then the class of all C -closed structures in Forb E ( E ) has Ramsey lift. J. Hubiˇ cka Ramsey Classes by Partite Construction I

  61. Unary closures = relations with out-degree 1 Unary closure description C is a set of pairs ( R U , R B ) where R U is unary relation and R B is binary relation. We say that structure A is C -closed if for every pair ( R U , R B ) the B -outdegree of every vertex of A that is in U is 1. Theorem (H., Nešetˇ ril, 2015) Let E be a family of complete ordered structures and U an unary closure description. Then the class of all C -closed structures in Forb E ( E ) has Ramsey lift. All Cherlin Shelah Shi classes with unary closure can be described this way! J. Hubiˇ cka Ramsey Classes by Partite Construction I

  62. The Induced Partite Construction with unary closure → ( B ) A ril-Rödl Theorem: C 0 − Nešetˇ 2 Construct C 0 -partite P 0 J. Hubiˇ cka Ramsey Classes by Partite Construction I

  63. The Induced Partite Construction with unary closure → ( B ) A ril-Rödl Theorem: C 0 − Nešetˇ 2 Construct C 0 -partite P 0 Enumerate by A 1 , . . . , A N all possible projections of copies of A in P 0 J. Hubiˇ cka Ramsey Classes by Partite Construction I

  64. The Induced Partite Construction with unary closure → ( B ) A ril-Rödl Theorem: C 0 − Nešetˇ 2 Construct C 0 -partite P 0 Enumerate by A 1 , . . . , A N all possible projections of copies of A in P 0 Construct C 0 -partite P 1 , . . . , P N : B i : partite system induced on P i − 1 by all copies of all with projection to A i → ( B i ) A i Partite lemma: C i − 2 P i is built by repeated free amalgamation of P i over all copies of B i in C i J. Hubiˇ cka Ramsey Classes by Partite Construction I

  65. The Induced Partite Construction with unary closure → ( B ) A ril-Rödl Theorem: C 0 − Nešetˇ 2 Construct C 0 -partite P 0 Enumerate by A 1 , . . . , A N all possible projections of copies of A in P 0 Construct C 0 -partite P 1 , . . . , P N : B i : partite system induced on P i − 1 by all copies of all with projection to A i → ( B i ) A i Partite lemma: C i − 2 P i is built by repeated free amalgamation of P i over all copies of B i in C i Put C = P N J. Hubiˇ cka Ramsey Classes by Partite Construction I

  66. The Induced Partite Construction with unary closure → ( B ) A ril-Rödl Theorem: C 0 − Nešetˇ 2 Construct C 0 -partite P 0 Enumerate by A 1 , . . . , A N all possible projections of copies of A in P 0 Construct C 0 -partite P 1 , . . . , P N : B i : partite system induced on P i − 1 by all copies of all with projection to A i → ( B i ) A i Partite lemma: C i − 2 P i is built by repeated free amalgamation of P i over all copies of B i in C i Put C = P N A , B , B i are C -closed. Only potential problem is the partite construction. J. Hubiˇ cka Ramsey Classes by Partite Construction I

  67. The Partite Lemma Easy description of C : X 1 a b B A = B = X 2 x y z B J. Hubiˇ cka Ramsey Classes by Partite Construction I

  68. The Partite Lemma Easy description of C : X 1 a b B A = B = X 2 x y z B Vertices in partition X i C : Functions f : { 1 , 2 , . . . , N } → X i B Add as many tuples to relations as possible such that all the evaluation maps g i ( f ) = f ( i ) are homomorphisms from C to B . J. Hubiˇ cka Ramsey Classes by Partite Construction I

  69. The Partite Lemma Easy description of C : X 1 a b B A = B = X 2 x y z B Vertices in partition X i C : Functions f : { 1 , 2 , . . . , N } → X i B Add as many tuples to relations as possible such that all the evaluation maps g i ( f ) = f ( i ) are homomorphisms from C to B . out-degree 1 is preserved: J. Hubiˇ cka Ramsey Classes by Partite Construction I

  70. The Partite Lemma Easy description of C : X 1 a b B A = B = X 2 x y z B Vertices in partition X i C : Functions f : { 1 , 2 , . . . , N } → X i B Add as many tuples to relations as possible such that all the evaluation maps g i ( f ) = f ( i ) are homomorphisms from C to B . out-degree 1 is preserved: f ( 1 ) = x , f ( 2 ) = y , f ( 3 ) = z , f ( 4 ) = x , . . . J. Hubiˇ cka Ramsey Classes by Partite Construction I

Download Presentation
Download Policy: The content available on the website is offered to you 'AS IS' for your personal information and use only. It cannot be commercialized, licensed, or distributed on other websites without prior consent from the author. To download a presentation, simply click this link. If you encounter any difficulties during the download process, it's possible that the publisher has removed the file from their server.

Recommend

Ramsey Classes by Partite Construction II Honza Hubi cka Mathematics and Statistics

Ramsey Classes by Partite Construction II Honza Hubi cka Mathematics and Statistics

Ramsey Classes by Partite Construction II Honza Hubi cka Mathematics and Statistics University of Calgary Calgary Institute of Computer Science Charles University Prague Joint work with Jaroslav Neet ril Permutation Groups and

1.12k views • 97 slides
All those Ramsey classes Ramsey classes with closures and forbidden homomorphisms Jan Hubi cka

All those Ramsey classes Ramsey classes with closures and forbidden homomorphisms Jan Hubi cka

Ramsey Structural Ramsey Ramsey classes Ramsey classes with constraints All those Ramsey classes Ramsey classes with closures and forbidden homomorphisms Jan Hubi cka Computer Science Institute of Charles University Charles University

1.32k views • 97 slides
Expanding Fra ss e classes into Ramsey classes L. Nguyen Van Th e (joint with Y. Gutman

Expanding Fra ss e classes into Ramsey classes L. Nguyen Van Th e (joint with Y. Gutman

Expanding Fra ss e classes into Ramsey classes L. Nguyen Van Th e (joint with Y. Gutman and T. Tsankov) Universit dAix-Marseille July 2012 L. Nguyen Van Th e (Aix-Marseille) Expanding Fra ss e classes into Ramsey

387 views • 15 slides
Ramsey theorems for classes of structures with functions and relations Jan Hubi cka

Ramsey theorems for classes of structures with functions and relations Jan Hubi cka

Structural Ramsey Free amalgamation classes Amalgamation with closures General amalgamation Open problems Ramsey theorems for classes of structures with functions and relations Jan Hubi cka Department of Applied Mathematics Charles

1.02k views • 98 slides
Ramsey classes Topological dynamics Model theory Ramsey theory The KPT theory Questions

Ramsey classes Topological dynamics Model theory Ramsey theory The KPT theory Questions

Introduction Questions Results Ramsey classes Miodrag Soki c Introduction Motivation Ramsey classes Topological dynamics Model theory Ramsey theory The KPT theory Questions Miodrag Soki c Posets Ultrametric spaces Caltech

831 views • 65 slides
Ramsey properties of Hrushovski construction Jan Hubi cka Computer Science Institute of

Ramsey properties of Hrushovski construction Jan Hubi cka Computer Science Institute of

Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Ramsey properties of Hrushovski construction Jan Hubi cka Computer Science Institute of Charles University Charles University Prague

1.97k views • 121 slides
RAMSEY CLASSES SPARSITY AND MODELS FOR FINITE - NESIETPIIL JAROSLAV UNIVERSITY CHARLES

RAMSEY CLASSES SPARSITY AND MODELS FOR FINITE - NESIETPIIL JAROSLAV UNIVERSITY CHARLES

RAMSEY CLASSES SPARSITY AND MODELS FOR FINITE - NESIETPIIL JAROSLAV UNIVERSITY CHARLES PRAGUE WITH MENDEZ DE PATRICE OSSONA , HUBIEKA JAN EVANS DAVID , _ JAN 31,2018 PARIS IHP , 1. CONTENTS SPARSITY & STABILITY

543 views • 32 slides
Student Engagement in Large Lecture Classes Jeremy Ramsey LITT Workshop Presentation Lycoming

Student Engagement in Large Lecture Classes Jeremy Ramsey LITT Workshop Presentation Lycoming

Using Internet Video To Encourage Student Engagement in Large Lecture Classes Jeremy Ramsey LITT Workshop Presentation Lycoming College September 16, 2014 Previous Work Students in CHEM 110 General Chemistry asked for more in-class

619 views • 8 slides
Automorphism Groups and Ramsey Properties of Sparse Graphs Jan Hubi cka Computer Science

Automorphism Groups and Ramsey Properties of Sparse Graphs Jan Hubi cka Computer Science

Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Automorphism Groups and Ramsey Properties of Sparse Graphs Jan Hubi cka Computer Science Institute of Charles University Charles

1.26k views • 115 slides
Ramsey classes and partial orders Anja Komatar Department of Pure Mathematics, University of

Ramsey classes and partial orders Anja Komatar Department of Pure Mathematics, University of

Ramsey classes and partial orders Anja Komatar Department of Pure Mathematics, University of Leeds Logic Colloquium, 2018 Anja Komatar (University of Leeds) Ramsey classes and partial orders Logic Colloquium, 2018 1 / 22 Outline Structural

371 views • 8 slides
Graph Generation and Ramsey Numbers Brendan McKay Vigleik Angeltveit Australian National

Graph Generation and Ramsey Numbers Brendan McKay Vigleik Angeltveit Australian National

Graph Generation and Ramsey Numbers Brendan McKay Vigleik Angeltveit Australian National University Ramsey Numbers 2017 1 Ramsey Theory Total disorder is impossible! Frank Plumpton Ramsey (19031930) Ramsey Numbers 2017 2 Ramsey

1.2k views • 109 slides
2 in -models Benoit Monin joint work with Ludovic Patey Universit e Paris-Est Cr eteil

2 in -models Benoit Monin joint work with Ludovic Patey Universit e Paris-Est Cr eteil

SRT 2 2 vs RT 2 2 in -models Benoit Monin joint work with Ludovic Patey Universit e Paris-Est Cr eteil NUS - IMS . Ramsey Theory Section 1 Ramsey Theory Controlling 0 Ramsey Theory Partition regular classes 2 statements

521 views • 51 slides
From abstract -Ramsey theory to abstract ultra-Ramsey Theory Timothy Trujillo SE OP

From abstract -Ramsey theory to abstract ultra-Ramsey Theory Timothy Trujillo SE OP

From abstract -Ramsey theory to abstract ultra-Ramsey Theory Timothy Trujillo SE OP 2016 Iriki Venac, Fruka gora Trujillo Abstract alpha-Ramsey Theory 1/22 Overview 1 Framework for the results Trujillo Abstract alpha-Ramsey

947 views • 68 slides
A Presentation to the Ramsey County Board of Commissioners www.headinghomeramsey.org 1 OUTLINE

A Presentation to the Ramsey County Board of Commissioners www.headinghomeramsey.org 1 OUTLINE

A Presentation to the Ramsey County Board of Commissioners www.headinghomeramsey.org 1 OUTLINE 1. What does homelessness look like in Ramsey? 2. What is Heading Home Ramsey? 3. What is Heading Home Ramsey working on? 4. How can Ramsey

559 views • 27 slides
Topological Ramsey Theory and the Rationals Stevo Todorcevic Bertinoro, May 23, 2011 Outline 1.

Topological Ramsey Theory and the Rationals Stevo Todorcevic Bertinoro, May 23, 2011 Outline 1.

Topological Ramsey Theory and the Rationals Stevo Todorcevic Bertinoro, May 23, 2011 Outline 1. Ramsey-classification theory 2. Topological Ramsey spaces 3. The Halpern-L auchli space of strong subtrees 4. Ramsey theory of the countable

1.59k views • 133 slides
An abstract approach to finite Ramsey theory and a self-dual Ramsey theorem S lawomir Solecki

An abstract approach to finite Ramsey theory and a self-dual Ramsey theorem S lawomir Solecki

An abstract approach to finite Ramsey theory and a self-dual Ramsey theorem S lawomir Solecki University of Illinois at UrbanaChampaign May 2011 Outline Outline of Topics Self-dual Ramsey theorem 1 Algebraic notions 2 Abstract

1.35k views • 132 slides
On the Images of Sobolev space under Schrodinger semigroup associated to the Dunkl operator

On the Images of Sobolev space under Schrodinger semigroup associated to the Dunkl operator

On the Images of Sobolev space under Schrodinger semigroup associated to the Dunkl operator Sivaramakrishnan C IIT Hyderabad 6 th Worshop on Fourier analysis and related fields, University of pecs. Joint work with Venku Naidu and Sukumar D

556 views • 39 slides
On lower bounds for C 0 -semigroups Yuri Tomilov IM PAN, Warsaw Chemnitz, August, 2017 Yuri

On lower bounds for C 0 -semigroups Yuri Tomilov IM PAN, Warsaw Chemnitz, August, 2017 Yuri

On lower bounds for C 0 -semigroups Yuri Tomilov IM PAN, Warsaw Chemnitz, August, 2017 Yuri Tomilov (IM PAN, Warsaw) On lower bounds for C 0 -semigroups Chemnitz, August, 2017 1 / 17 Trivial bounds For f

1.08k views • 56 slides
Some nilpotent and commutative finite semigroups Manuel Delgado WORKSHOP ON SEMIGROUPS (To

Some nilpotent and commutative finite semigroups Manuel Delgado WORKSHOP ON SEMIGROUPS (To

Some nilpotent and commutative finite semigroups Manuel Delgado WORKSHOP ON SEMIGROUPS (To remember John M. Howie on the occasion of his 76th birthday) Lisbon, 23-25 May, 2012 Numerical Semigroups Quotients Computations Nilpotent semigroups

269 views • 14 slides
Universal constraints on semigroups of hyperbolic isometries Argyris Christodoulou The Open

Universal constraints on semigroups of hyperbolic isometries Argyris Christodoulou The Open

Universal constraints on semigroups of hyperbolic isometries Argyris Christodoulou The Open University CAFT 2018 Problem and Motivation Determine all the finite collections of hyperbolic isometries f 1 f 2 f n

614 views • 44 slides
Generating functionals on quantum groups Adam Skalski 1 Ami Viselter ,2 1 IMPAN, Warsaw 2

Generating functionals on quantum groups Adam Skalski 1 Ami Viselter ,2 1 IMPAN, Warsaw 2

Generating functionals on quantum groups Adam Skalski 1 Ami Viselter ,2 1 IMPAN, Warsaw 2 University of Haifa Quantum Groups and their Analysis University of Oslo August 8, 2019 Ami Viselter (University of Haifa) Generating functionals on

1.42k views • 93 slides
Riesz transforms on group von Neumann algebras Tao Mei Wayne State University Wuhan Joint work

Riesz transforms on group von Neumann algebras Tao Mei Wayne State University Wuhan Joint work

Riesz transforms on group von Neumann algebras Tao Mei Wayne State University Wuhan Joint work with M. Junge and J. Parcet June 17, 2014 Classical Riesz transform f L 2 ( R n ), R = 1 2 . with ) , = 2 2

203 views • 16 slides
Idempotent generation in partition monoids James East University of Western Sydney Workshop on

Idempotent generation in partition monoids James East University of Western Sydney Workshop on

Idempotent generation in partition monoids James East University of Western Sydney Workshop on Diagram Algebras 812 Sept 2014 Universit at Stuttgart James East Idempotent generation in partition monoids Joint work with Bob Gray (and

1.7k views • 134 slides
Affine Processes are regular Martin Keller-Ressel kemartin@math.ethz.ch ETH Z urich Based on

Affine Processes are regular Martin Keller-Ressel kemartin@math.ethz.ch ETH Z urich Based on

Affine Processes are regular Martin Keller-Ressel kemartin@math.ethz.ch ETH Z urich Based on joint work with Walter Schachermayer and Josef Teichmann (arXiv:0906.3392) Conference on Analysis, Stochastics, and Applications, Vienna, July 12,

742 views • 25 slides

More recommend