Gallai–Ramsey Numbers for Rainbow Paths

Given graphs G and H and a positive integer k, the Gallai–Ramsey number, denoted by gr_k(G: H) is defined to be the minimum integer n such that every coloring of K_n using at most k colors will contain either a rainbow copy of G or a monochromatic copy of H. We consider this question in the cases where G∈ { P₄, P₅}. In the case where G= P₄, we completely solve the Gallai–Ramsey question by reducing to the 2-color Ramsey numbers. In the case where G= P₅, we conjecture that the problem reduces to the 3-color Ramsey numbers and provide several results in support of this conjecture.