Ramsey and Gallai-Ramsey numbers for linear forests and kipas

For two graphs G,H, the Ramsey number r(G,H) is the minimum integer n such that any red/blue edge-coloring of K_n contains either a red copy of G or a blue copy of H. For two graphs G,H, the Gallai-Ramsey number gr_k(G:H) is defined as the minimum integer n such that any k-edge-coloring of K_n must contain either a rainbow copy of G or a monochromatic copy of H. In this paper, the classical Ramsey numbers of linear forest versus kipas are obtained. We obtain the exact values of gr_k(G:H), where H is either a path or a kipas and G∈{K_1,3,P_4^+,P_5} and P_4^+ is the graph consisting of P_4 with one extra edge incident with inner vertex.