Gallai-Ramsey number for K5

Given a graph H, the k-colored Gallai-Ramsey number gr(k)(K-3 : H) is defined to be the minimum integer n such that every k-coloring of the edges of the complete graph on n vertices contains either a rainbow triangle or a monochromatic copy of H. 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 H = K-4. In this paper we attack the next case, when H = K-5. Surprisingly it turns out, that the validity of the conjecture depends upon the (yet unknown) value of the Ramsey number R(5, 5). It is known that 43 <= R(5, 5) <= 48 and conjectured that R(5, 5) = 43. If 44 <= R(5, 5) <= 48, then Fox et al.'s conjecture is true and we present a complete proof. If, however, R(5, 5) = 43, 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 R(5, 5) = 43, we show lower and upper bounds for the Gallai-Ramsey number gr(k)(K-3 : K-5).