Rainbow Turán Problems

Source abstract

For a fixed graph $H$ , we define the rainbow Turán number $\ex^*(n,H)$ to be the maximum number of edges in a graph on $n$ vertices that has a proper edge-colouring with no rainbow $H$ . Recall that the (ordinary) Turán number $\ex(n,H)$ is the maximum number of edges in a graph on $n$ vertices that does not contain a copy of $H$ . For any non-bipartite $H$ we show that $\ex^*(n,H)=(1+o(1))\ex(n,H)$ , and if $H$ is colour-critical we show that $\ex^{*}(n,H)=\ex(n,H)$ . When $H$ is the complete bipartite graph $K_{s,t}$ with $s \leq t$ we show $\ex^*(n,K_{s,t}) = O(n^{2-1/s})$ , which matches the known bounds for $\ex(n,K_{s,t})$ up to a constant. We also study the rainbow Turán problem for even cycles, and in particular prove the bound $\ex^*(n,C_6) = O(n^{4/3})$ , which is of the correct order of magnitude.