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 [Formula presented] 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(K₃:K_1,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 egr_k(S₃⁺:K_1,t) for all integers t≥1 and k≥3t−2. We also determine egr_k(S₃⁺:K_1,t) for integers 1≤t≤5 and k≥4.