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...

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 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...

We call a subgraph of an edge-colored graph rainbow, if all its edges have different colors. The anti-Ramsey number G in complete K n , denoted by r ( ) is the maximum colors edge-coloring with no rainbow copy . In this paper, we determine exact value for star forests and double stars approximate linear forests.