Monochromatic stars in rainbow [formula omitted]-free and [formula omitted]-free colorings

Abstract

Given two graphs G and H, we consider the Ramsey-type problem of finding the minimum integer n (denoted by egrk(G:H)) such that n2≥k and for every N≥n, every rainbow G-free k-coloring (using exactly k colors) of the complete graph KN contains a monochromatic copy of H. In this paper, we determine egrk(K3:K1,t) for all integers t≥1 and k≥3 completely. Let S3+ be the unique graph on four vertices consisting of a triangle and a pendant edge. We characterize egrk(S3+:K1,t) for all integers t≥1 and k≥3t−2. We also determine egrk(S3+:K1,t) for integers 1≤t≤5 and k≥4.