Gallai–Ramsey number for K5 ${K}_{5}$

Abstract Given a graph , the ‐colored Gallai–Ramsey number is defined to be the minimum integer such that every ‐coloring of the edges of the complete graph on vertices contains either a rainbow triangle or a monochromatic copy of Fox et al. conjectured the values of the Gallai–Ramsey numbers for complete graphs. Recently, this conjecture has been verified for the first open case, when . In this paper we attack the next case, when . Surprisingly it turns out, that the validity of the conjecture depends upon the (yet unknown) value of the Ramsey number . It is known that and conjectured that . If , then Fox et al.'s conjecture is true and we present a complete proof. If, however, , then Fox et al.'s conjecture is false, meaning that exactly one of these conjectures is true while the other is false. For the case when , we show lower and upper bounds for the Gallai–Ramsey number .