This is a question I've implicitly asked on AoPS five years ago, and has not been answered. I apologize for its possible simplicity, as I am not a graph theorist.

Let $n$ and $k$ be two positive integers. Let $G$ be a graph with vertex set $V$. Assume that $V$ is partitioned into $k$ disjoint subsets $V_1, V_2, \ldots, V_k$, each of which has exactly $n$ elements (so that $\left|V\right| = nk$). Assume that for each $i \in \left\{1, 2, \ldots, k\right\}$, the set $V_i$ is an independent set of $G$ (that is, no two vertices of $V_i$ are connected by an edge). Assume also that each vertex of $G$ has degree $\geq \left(k-2\right)n+1$.

Prove or disprove that $G$ has a $k$-clique (that is, $k$ distinct vertices pairwise connected by edges).

This is a triviality when $k = 1$ or $k = 2$, and a known contest problem in the case when $k = 3$ (see http://artofproblemsolving.com/community/c6h40982 , and http://artofproblemsolving.com/community/c6h49347 for an extension which is not completely true as stated). In my AoPS thread linked above, I have asked whether the $k = 4$ case is true, but of course the real question is the general case.