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

Given two graphs G and H, we consider the Ramsey-type problem of finding the minimum integer n (denoted by egr(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 egr(k)(K-3 : K-1,K- 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 egr(k)(S-3(+) 3 : K-1,K- t) for all integers t >= 1 and k >= 3t - 2. We also determine egr(k)(S-3(+) : K-1,K- t) for integers 1 <= t <= 5 and k >= 4. (c) 2020 Elsevier B.V. All rights reserved.