  A Gallai-coloring (Gallai-$k$-coloring) is an edge-coloring (with colors from
$\{1, 2, \ldots, k\}$) of a complete graph without rainbow triangles. Given a
graph $H$ and a positive integer $k$, the $k$-colored Gallai-Ramsey number
$GR_k(H)$ is the minimum integer $n$ such that every Gallai-$k$-coloring of the
complete graph $K_n$ contains a monochromatic copy of $H$. In this paper, we
consider two extremal problems related to Gallai-$k$-colorings. First, we
determine upper and lower bounds for the maximum number of edges that are not
contained in any rainbow triangle or monochromatic triangle in a
$k$-edge-coloring of $K_n$. Second, for $n\geq GR_k(K_3)$, we determine upper
and lower bounds for the minimum number of monochromatic triangles in a
Gallai-$k$-coloring of $K_{n}$, yielding the exact value for $k=3$.
Furthermore, we determine the Gallai-Ramsey number $GR_k(K_4+e)$ for the graph
on five vertices consisting of a $K_4$ with a pendant edge.
