Choose $k$. Let $G = (V,E)$ be a graph on $n = R(k,k)-1$ vertices (that is, $G$ is an extremal example for $R(k,k)$, and $g : E \to \{r, b\}$ be an edge 2-coloring such that there is no monochromatic $k$ clique.

Define $\tau(i)$ to be the proportion of $K_k \subset G$ such that exactly $i$ of the edges are colored $r$.

Questions: Are there any (nontrivial) properties known about $\tau$? Most interestingly, does $\tau \to$ normal distribution as $k \to \infty$?