  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. [J. Fox, A. Grinshpun, and J. Pach.
The Erd{\H o}s-Hajnal conjecture for rainbow triangles. J. Combin. Theory Ser.
B, 111:75-125, 2015.] conjectured the value 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 $43 \leq R(5,5) \leq 48$ and
conjectured that $R(5,5)=43$ [B.D. McKay and S.P. Radziszowski. Subgraph
counting identities and Ramsey numbers. J. Combin. Theory Ser. B, 69:193-209,
1997]. If $44 \leq R(5,5) \leq 48$, 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 at least one of these two conjectures must be
false. For the case when $R(5, 5) = 43$, we show lower and upper bounds for the
Gallai Ramsey number $gr_{k}(K_{3} : K_5)$.
