Let H s → G mean that every s-coloring of E(H) produces a monochromatic copy of G in some color class. Let the s-color degree Ramsey number of a graph G, written R∆(G; s), be min{∆(H) : H s → G}. We prove that the 2-color degree Ramsey number is at most 96 for every even cycle and at most 3458 for every odd cycle. For the general s-color problem on even cycles, we prove R∆(C2m; s) ≤ 16s6 for al...

Let Cn denote the 3-uniform hypergraph loose cycle, that is the hypergraph with vertices v1, . . . , vn and edges v1v2v3, v3v4v5, v5v6v7, . . . , vn−1vnv1. We prove that every red-blue colouring of the edges of the complete 3-uniform hypergraph with N vertices contains a monochromatic copy of Cn, where N is asymptotically equal to 5n/4. Moreover this result is (asymptotically) best possible.

In this paper, we study Ramsey-type problems for directed graphs. We first consider the k-colour oriented Ramsey number of H, denoted by − →r (H, k), which is the least n for which every k-edgecoloured tournament on n vertices contains a monochromatic copy of H. We prove that − →r (T, k) ≤ ck|T |k for any oriented tree T . This is a generalisation of a similar result for directed paths by Chvát...

‎For a set of non-negative integers~$L$‎, ‎the $L$-intersection number of a graph is the smallest number~$l$ for which there is an assignment of subsets $A_v subseteq {1,dots‎, ‎l}$ to vertices $v$‎, ‎such that every two vertices $u,v$ are adjacent if and only if $|A_u cap A_v|in L$‎. ‎The bipartite $L$-intersection number is defined similarly when the conditions are considered only for the ver...

There are two equivalent problem statements for the Ramsey number n = R(k; l). n is the minimum number of vertices in the graph such that it contains a complete graph of k vertices, or an independent set of l vertices. n is the minimum number of vertices such that if all the edges of the complete graph on n vertices, denoted by K n is colored with two colors, fRed, Blueg, then there exists a Re...

Let P(x,y,n) be a real polynomial and let {G, } be a family of graphs, where the set of vertices of G, is (1.2,.. . ,n} and for 1 I i < j' 5 n {;,I] is an edge of G, iff P(i,j,n) > 0. Motivated by a question of Babai, we show that there is a positiye constant c depending only on P such that either G or its complement G, contains _a complete subgraph on at least c2 vertices. Similarly, either G,...

Let G and H be simple graphs. The Ramsey number r(G,H) for a pair of graphs G and H is the smallest number r such that any red-blue coloring of the edges of Kr contains a red subgraph G or a blue subgraph H . The size Ramsey number r̂(G,H) for a pair of graphs G and H is the smallest number r̂ such that there exists a graph F with size r̂ satisfying the property that any red-blue coloring of the e...