On Size Bipartite and Tripartite Ramsey Numbers for The Star Forest and Path on 3 Vertices

Abstract

For simple graphs G and H the size multipartite Ramsey number mj ( G , H ) is the smallest natural number t such that any arbitrary red-blue coloring on the edges of Kjxt contains a red G or a blue H as a subgraph. We studied the size tripartite Ramsey numbers m 3( G , H ) where G=mK1,n  and H=P3 . In this paper, we generalize this result. We determine m3(G,H) where G is a star forest, namely a disjoint union of heterogeneous stars, and H=P3 . Moreover, we also determine m2(G,H) for this pair of graphs G and H .