A Size Multipartite Ramsey Problem Involving the Claw Graph

Abstract

Let K j×s denote a complete balanced multipartite graph consisting of j partite sets of uniform size s. For any two colouring of the edges of a graph K j×s , we say that K j×s → (K 1,3 ,G), if there exists a copy of K 1,3 (Claw graph) in the first colour or a copy of G in the second colour. m j (K 1,3 ,G) is defined as the smallest positive integer s such that K j×s → (K 1,3 ,G). In this paper we find all such m j (K 1,3 ,G) for all graphs G on 4 vertices.