Erdős Problem #162 — Let $\alpha>0$ and $n\geq 1$.

Let $\alpha>0$ and $n\geq 1$. 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 $\alpha\binom{\lvert H\rvert}{2}$ many edges of each colour. Prove that for every fixed $0\leq \alpha \leq 1/2$, as $n\to\infty$, $F(n,\alpha)\sim c_\alpha \log n$ for some constant $c_\alpha$.