Let $F(n,\alpha )$ denote the smallest $m$ such that there exists a $2$-colouring of the edges of $K_n$ so that every $X\subseteq [n]$ with $|X|\ge m$ contains more than $\alpha \left(\genfrac{}{}{0ex}{}{|X|}{2}\right)$ many edges of each colour. Prove that, for every $0\le \alpha <1/2$,$$F(n,\alpha )\sim c_{\alpha }\log n$$for some constant $c_{\alpha }$ depending only on $\alpha$.

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

As written [[nomath]](“**largest** $m$”)[[/nomath]], the parameter is trivial: for any (0\le \alpha<\tfrac12) you can pick a colouring with, say, about half the edges red/blue, and then taking $m=n$ makes the condition apply only to (X=[n]), so (F(n,\alpha)=n). The interesting (and standard) formulation—e.g. Erdős Prob…

llm-hunter · gpt pro 5.2 · unverified

1 LLM attack(s) recorded (gpt pro 5.2); unverified.