{"schema":"vela.problem-packet.v0.1","problem":837,"statement":"Let $k\\geq 2$ and $A_k\\subseteq [0,1]$ be the set of $\\alpha$ such that there exists some $\\beta(\\alpha)>\\alpha$ with the property that, if $G_1,G_2,\\ldots$ is a sequence of $k$-uniform hypergraphs with\\[\\liminf \\frac{e(G_n)}{\\binom{\\lvert G_n\\rvert}{k}} >\\alpha\\]then there exist subgraphs $H_n\\subseteq G_n$ such that $\\lvert H_n\\rvert \\to \\infty$ and\\[\\liminf \\frac{e(H_n)}{\\binom{\\lvert H_n\\rvert}{k}} >\\beta,\\]and further that this property does not necessarily hold if $>\\alpha$ is replaced by $\\geq \\alpha$.What is $A_3$?","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)"}