  For fixed $p$ and $q$, an edge-coloring of the complete graph $K_n$ is said
to be a $(p, q)$-coloring if every $K_p$ receives at least $q$ distinct colors.
The function $f(n, p, q)$ is the minimum number of colors needed for $K_n$ to
have a $(p, q)$-coloring. This function was introduced about 45 years ago, but
was studied systematically by Erd\H{o}s and Gy\'{a}rf\'{a}s in 1997, and is now
known as the Erd\H{o}s-Gy\'{a}rf\'{a}s function. In this paper, we study $f(n,
p, q)$ with respect to Gallai-colorings, where a Gallai-coloring is an
edge-coloring of $K_n$ without rainbow triangles. Combining the two concepts,
we consider the function $g(n, p, q)$ that is the minimum number of colors
needed for a Gallai-$(p, q)$-coloring of $K_n$. Using the anti-Ramsey number
for $K_3$, we have that $g(n, p, q)$ is nontrivial only for $2\leq q\leq p-1$.
We give a general lower bound for this function and we study how this function
falls off from being equal to $n-1$ when $q=p-1$ and $p\geq 4$ to being
$\Theta(\log n)$ when $q = 2$. In particular, for appropriate $p$ and $n$, we
prove that $g=n-c$ when $q=p-c$ and $c\in \{1,2\}$, $g$ is at most a fractional
power of $n$ when $q=\lfloor\sqrt{p-1}\rfloor$, and $g$ is logarithmic in $n$
when $2\leq q\leq \lfloor\log_2 (p-1)\rfloor+1$.
