Solved by Erdős, Rousseau, and Schelp for large $n$, but unpublished. Alon has observed that this also follows from a result of Pyber [Py86], which states that (for large enough $n$) at most $\lfloor n^2/4\rfloor+2$ monochromatic cliques cover all edges of a $2$-coloured $K_n$.

This problem was solved completely by Keevash and Sudakov [KeSu04], who provd that the corret threshold is $\lfloor n^2/4\rfloor$ for all $n\geq 7$, is $\binom{n}{2}$ for $n\leq 5$, and is $10$ for $n=6$.