Erdős problem 639

Is it true that if the edges of $K_n$ are 2-coloured then there are at most $n^2/4$ many edges which do not occur in a monochromatic triangle?

Solved by Erdős, Rousseau, and Schelp for large $n$, but unpublished.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 proved that the correct threshold is $\lfloor n^2/4\rfloor$ for all $n\geq 7$, is $\binom{n}{2}$ for $n\leq 5$, and is $10$ for $n=6$.

Since the bound fails for small $n$ (at $n=6$ the threshold is $10 > 6^2/4$), the statement is formalized in the asymptotic reading in which the problem was posed and solved: for all sufficiently large $n$, every $2$-colouring of the edges of $K_n$ leaves at most $n^2/4$ edges not occurring in a monochromatic triangle. Edges of $K_n$ are the non-diagonal unordered pairs Sym2 (Fin n); an edge $\{x, y\}$ occurs in a monochromatic triangle if and only if there is a third vertex $z$ with $C(\{x, z\}) = C(\{y, z\}) = C(\{x, y\})$.

The linked file proves the bound for every finite vertex type with at least $10$ vertices, which gives the atTop reading below, and collects the uncovered edges as the edge set of a graph rather than as a set of pairs.