{"schema":"vela.problem-packet.v0.1","problem":162,"statement":"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 $c_\\alpha$.","status":"open","seam":"raw","closureRoutes":[],"obligations":[],"attestations":[],"attempts":[],"velaLean":[],"oeis":[],"generated_note":"derived view; the signed event log is the source of truth (vela check / vela reproduce)"}