Let $\alpha >0$ and $n\ge 1$. Let $F(n,\alpha )$ be the largest $k$ such that there exists some 2-colouring of the edges of $K_n$ in which any induced subgraph $H$ on at least $k$ vertices contains more than $\alpha \left(\genfrac{}{}{0ex}{}{|H|}{2}\right)$ many edges of each colour.Prove that for every fixed $0\le \alpha \le 1/2$, as $n\to \infty$,$$F(n,\alpha )\sim c_{\alpha }\log n$$for some constant $c_{\alpha }$.