Monochromatic stars in rainbow K3-free and S3+-free colorings

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