Given a graph H, the size Ramsey number re(H, q) is the minimal number m for which there is a graph G with m edges such that every q-coloring of G contains a monochromatic copy of H. We study the size Ramsey number of the directed path of length n in oriented graphs, where no antiparallel edges are allowed. We give nearly tight bounds for every fixed number of colors, showing that for every q ≥...

In this paper we study multipartite Ramsey numbers for odd cycles. Our main result is the proof of a conjecture of Gyárfás, Sárközy and Schelp [12]. Precisely, let n ≥ 5 be an arbitrary positive odd integer; then in any two-coloring of the edges of the complete 5-partite graph K(n−1)/2,(n−1)/2,(n−1)/2,(n−1)/2,1 there is a monochromatic cycle of length n. keywords: cycles, Ramsey number, Regular...

Let c(m, zi) be the least integer p such that, for any graph G of order p, either G has an zzi-cycle or its complement G has an zz-cycle. Values of c(m, n) are established for zzz, zi < 6 and general formulas are proved for c(3, zi), c(4, n), and c(5, zz). Introduction. It is a well-known fact that in any gathering of six people, there are three people who are mutual acquaintances or three peop...

If G and H are graphs, let us write G → (H)2 if G contains a monochromatic copy ofH in any 2-colouring of the edges of G. The size-Ramsey number re(H) of a graph H is the smallest possible number of edges a graph G may have if G→ (H)2. Suppose T is a tree of order |T | ≥ 2, and let t0, t1 be the cardinalities of the vertex classes of T as a bipartite graph, and let ∆(T ) be the maximal degree o...

The notion of a graph theoretic Ramsey number is generalised by assuming that both the original graph whose edges are arbitrarily bi–coloured and the sought after monochromatic subgraphs are complete, balanced, multipartite graphs, instead of complete graphs as in the classical definition. We previously confined our attention to diagonal multipartite Ramsey numbers. In this paper the definition...

The graph Ramsey number R(G,H) is the smallest integer n such that every 2-coloring of the edges of Kn contains either a red copy of G or a blue copy of H . We find the largest star that can be removed from Kn such that the underlying graph is still forced to have a red G or a blue H . Thus, we introduce the star-avoiding Ramsey number r∗(G,H) as the smallest integer k such that every 2-colorin...