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 , and H is some graph with a red-blue coloring of its edges. How large can R and B be if R ∪ B does not contain a copy of H ? Call the largest such integer 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 mex ( n , H ) = ex ( n , K k + 1 ). This conjecture generalizes Turán's theorem. Diwan and Mubayi also asked for an analogue of Erdős-Stone-Simonovits theorem in this context. We prove the following upper bound on the extremal threshold in terms of the chromatic number χ ( H ) and the reduced maximum matching number M ( H ) of H . mex ( n , H ) ≤ ( 1 − 1 2 ( χ ( H ) − 1 ) − M ( H ) 9 χ ( H ) 2 ) n 2 2 . M ( H ) is, among the set of proper χ ( 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 mex ( n , H ) when H is a cycle with a red-blue coloring of its edges, and we show that mex ( n , H ) ≲ 1 2 ( n 2 ), which is tight.