On a colored Tur\'an problem of Diwan and Mubayi

Abstract

Suppose that $R$ (red) and $B$ (blue) are two graphs on the same vertex set of size $n$, and $H$ is some graph with a red-blue coloring of its edges. How large can $R$ and $B$ be if $R\cup B$ does not contain a copy of $H$? Call the largest such integer $\mathrm{mex}(n, H)$. This problem was introduced by Diwan and Mubayi, who conjectured that (except for a few specific exceptions) when $H$ is a complete graph on $k+1$ vertices with any coloring of its edges $\mathrm{mex}(n,H)=\mathrm{ex}(n, K_{k+1})$. This conjecture generalizes Tur\'an's theorem. Diwan and Mubayi also asked for an analogue of Erd\H{o}s-Stone-Simonovits theorem in this context. We prove the following asymptotic characterization of the extremal threshold in terms of the chromatic number $\chi(H)$ and the \textit{reduced maximum matching number} $\mathcal{M}(H)$ of $H$. $$\mathrm{mex}(n, H)=\left(1- \frac{1}{2(\chi(H)-1)} - \Omega\left(\frac{\mathcal{M}(H)}{\chi(H)^2}\right)\right)\frac{n^2}{2}.$$ $\mathcal{M}(H)$ is, among the set of proper $\chi(H)$-colorings of $H$, the largest set of disjoint pairs of color classes where each pair is connected by edges of just a single color. The result is also proved for more than $2$ colors and is tight up to the implied constant factor. We also study $\mathrm{mex}(n, H)$ when $H$ is a cycle with a red-blue coloring of its edges, and we show that $\mathrm{mex}(n, H)\lesssim \frac{1}{2}\binom{n}{2}$, which is tight.