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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

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