Erdős Problem 162

### What is the conjecture

https://www.erdosproblems.com/162

Let $\alpha>0$ and $n\geq 1$. Let $F(n,\alpha)$ be the largest $k$ such that in any 2-colouring of the edges of $K_n$ any 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 $c_\alpha$.

Status: open

### Choose either option
- [ ] I plan on working on this conjecture
- [x] This issue is up for grabs: I would like to see this conjecture added by somebody else