Rainbow Ramsey simple structures

European Journal of Combinatorics, 1985

A class J{ of relational systems (of the same type) has the 8-Ramsey property if for every $ E J{ there is T E J{ such that to every 2-coloring of (!) (= relational subsystems of T isomorphic to 8) we can find a monochromatic (iJ for some {J E (D. Extending recent results by JeZek and Ndetnl we prove it for (a) every class J{ of finite reflexive relational systems closed for products and 8 E J{ a singleton, (b) every abstract class J{ of finite relational systems with the strong amalgamation property and 8 E J{ such that the sets from (!) are disjoint for all $ E J{. Finally we prove: Let J{ be an abstract class of finite reflexive or areflexive relational systems with the strong amalgamation property. If J{ has the 8-Ramsey property, then 8 is constant.

Journal of Combinatorial Theory, 1999

In this paper, we introduce a measure of the extent to which a finite combinatorial structure is a Ramsey object in the class of objects with a similar structure. We show for classes of finite relational structures, including graphs, binary posets, and bipartite graphs, how this measure depends on the symmetries of the structure.

Journal of Combinatorial Theory, Series B, 2002

For a graph F and natural numbers a 1 ; . . . ; a r ; let F ! ða 1 ; . . . ; a r Þ denote the property that for each coloring of the edges of F with r colors, there exists i such that some copy of the complete graph K ai is colored with the ith color. Furthermore, we write ða 1 ; . . . ; a r Þ ! ðb 1 ; . . . ; b s Þ if for every F for which F ! ða 1 ; . . . ; a r Þ we have also F ! ðb 1 ; . . . ; b s Þ: In this note, we show that a trivial sufficient condition for the relation ða 1 ; . . . ; a r Þ ! ðb 1 ; . . . ; b s Þ is necessary as well. # 2002 Elsevier Science (USA) # 2002 Elsevier Science (USA)

In combinatorics, Ramsey Theory considers partitions of some mathematical objects and asks the following question: how large must the original object be in order to guarantee that at least one of the parts in the partition exhibits some property? Perhaps the most familiar case is the well-known Pigeonhole Principle: if m pigeonholes house p pigeons where p m, then one of the pigeonholes must contain multiple pigeons. Conversely, the number of pigeons must exceed m in order to guarantee this property.

2010

We show that the class of finite rooted binary plane trees is a Ramsey class (with respect to topological embeddings that map leaves to leaves). That is, for all such trees P, H and every natural number k there exists a tree T such that for every k-coloring of the (topological) copies of P in T there exists a (topological) copy H of H in T such that all copies of P in H have the same color. When the trees are represented by the so-called rooted triple relation, the result gives rise to a Ramsey class of relational structures with respect to induced substructures.

Graphs and Combinatorics

We prove induced Ramsey theorems in which the monochromatic induced subgraph satisfies that all members of a prescribed set of its partial isomorphisms extend to automorphisms of the colored graph (without requirement of preservation of colors).

Journal of Mathematical Logic

Hindman's theorem says that every finite coloring of the natural numbers has a monochromatic set of finite sums. Ramsey algebras are structures that satisfy an analogue of Hindman's theorem. This paper introduces Ramsey algebras and presents some elementary results. Furthermore, their connection to Ramsey spaces will be addressed.

Journal of the Brazilian …, 2001

As usual, for graphs Γ, G, and H, we write Γ → (G, H) to mean that any red-blue colouring of the edges of Γ contains a red copy of G or a blue copy of H. A pair of graphs (G, H) is said to be Ramsey-infinite if there are infinitely many minimal graphs Γ for which we have Γ → (G, H). Let ≥ 4 be an integer. We show that if H is a 2connected graph that does not contain induced cycles of length at least , then the pair (C k , H) is Ramseyinfinite for any k ≥ , where C k denotes the cycle of length k.

Discrete Mathematics, 1982