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 \(G \in \{P_{4}, P_{5}\}\). In the case where \(G = P_{4}\), we completely solve the Gallai–Ramsey question by reducing to the 2-color Ramsey numbers. In the case where \(G = P_{5}\), we conjecture that the problem reduces to the 3-color Ramsey numbers and provide several results in support of this conjecture.

给定图G和H以及正整数k，用\（gr_ {k}（G：H）\）表示的Gallai–Ramsey数被定义为最小整数n，使得\（K_ {n } \）使用最多k种颜色将包含G的彩虹副本或H的单色副本。我们在\（G \ in \ {P_ {4}，P_ {5} \} \）的情况下考虑这个问题。在\（G = P_ {4} \）的情况下，我们通过简化为2色Ramsey数来完全解决Gallai–Ramsey问题。在\（G = P_ {5} \）的情况下，我们猜想该问题将减少为3色Ramsey数，并提供若干结果支持该猜想。