Turán Numbers of Several Bipartite Graphs

For graphs \(H_1,H_2,\dots ,H_k\), the k-color Turán number \(ex(n,H_1,H_2,\dots ,H_k)\) is the maximum number of edges in a k-colored graph of order n that does not contain monochromatic \(H_i\) in color i as a subgraph, where \(1\le i\le k\). In this note, we show that if \(H_i\) is a bipartite graph with at least two edges for \(1\le i\le k\), then \(ex(n,H_1,H_2,\dots ,H_k)=(1+o(1))\sum _{i=1}^kex(n,H_i)\) as \(n\rightarrow \infty \), in which the non-constructive proof for some cases can be derandomized.