On a colored Turán problem of Diwan and Mubayi

Suppose that R (red) and B (blue) are two graphs on the same vertex set of size n, H is some graph with a red-blue coloring its edges. How large can be if R∪B does not contain copy H? Call largest such integer mex(n,H). This problem was introduced by Diwan Mubayi, who conjectured (except for few specific exceptions) when complete k+1 vertices any edges mex(n,H)=ex(n,Kk+1). conjecture generalizes Turán's theorem. Mubayi also asked an analogue Erdős-Stone-Simonovits theorem in this context. We prove following upper bound extremal threshold terms chromatic number χ(H) reduced maximum matching M(H) H.mex(n,H)≤(1−12(χ(H)−1)−M(H)9χ(H)2)n22. is, among proper χ(H)-colorings H, disjoint pairs color classes where each pair connected just single color. The result proved more than 2 colors tight up to implied constant factor. study mex(n,H) cycle edges, we show mex(n,H)≲12(n2), which tight.