Turán-Ramsey problems (Q1923521)

Given graphs \(F_1,\dots, F_k\), write \(\text{ex}_k(n;F_1,\dots,F_k)\) for the maximal size of a graph \(G\) of order \(n\) whose edges can be coloured with \(k\) colours such that \(G\) contains no \(F_i\) all of whose edges are coloured with the \(i\)th colour. Alternatively, \(\text{ex}_k(n; F_1,\dots,F_k)\) is the maximal size of \(G_1\cup\cdots\cup G_k\), where \(G_1,\dots, G_k\) are graphs on the same set of \(n\) vertices, and no \(G_i\) contains \(F_i\) as a subgraph. This note proves the asymptotic bound for this function, and discusses related problems.