  Alon, Balogh, Keevash and Sudakov proved that the $(k-1)$-partite Tur\'an
graph maximizes the number of distinct $r$-edge-colorings with no monochromatic
$K_k$ for all fixed $k$ and $r=2,3$, among all $n$-vertex graphs. In this
paper, we determine this function asymptotically for $r=2$ among $n$-vertex
graphs with sub-linear independence number. Somewhat surprisingly, unlike
Alon-Balogh-Keevash-Sudakov's result, the extremal construction from
Ramsey-Tur\'an theory, as a natural candidate, does not maximize the number of
distinct edge-colorings with no monochromatic cliques among all graphs with
sub-linear independence number, even in the 2-colored case.
  In the second problem, we determine the maximum number of triangles
asymptotically in an $n$-vertex $K_k$-free graph $G$ with $\alpha(G)=o(n)$. The
extremal graphs have similar structure to the extremal graphs for the classical
Ramsey-Tur\'an problem, i.e.~when the number of edges is maximized.
