{"schema":"vela.problem-packet.v0.1","problem":563,"statement":"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 $\\lvert X\\rvert\\geq m$ contains more than $\\alpha \\binom{\\lvert X\\rvert}{2}$ many edges of each colour. Prove that, for every $0\\leq \\alpha< 1/2$,\\[F(n,\\alpha)\\sim c_\\alpha\\log n\\]for some constant $c_\\alpha$ depending only on $\\alpha$.","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)"}