Vanishing codegree Turán density implies vanishing uniform Turán density

Abstract

For a $k$-uniform hypergraph (or simply $k$-graph) $F$, the codegree Turán density $π_{\mathrm{co}}(F)$ is the infimum over all $α$ such that any $n$-vertex $k$-graph $H$ with every $(k-1)$-subset of $V(H)$ contained in at least $αn$ edges has a copy of $F$. The uniform Turán density $π_{\therefore}(F)$ is the supremum over all $d$ such that there are infinitely many $F$-free $k$-graphs $H$ satisfying that any linear-size subhypergraph of $H$ has edge density at least $d$. Falgas-Ravry, Pikhurko, Vaughan and Volec [J. London Math. Soc., 2023] asked whether for every $3$-graph $F$, $π_{\therefore}(F)\leqπ_{\mathrm{co}}(F)$. We provide a positive answer to this question provided that $π_{\mathrm{co}}(F)=0$. Our proof relies on a random geometric construction and a new formulation of the characterization of $3$-graphs with vanishing uniform Turán density due to Reiher, R{ö}dl and Schacht [J. London Math. Soc., 2018]. Along the way, we answer a question of Falgas-Ravry, Pikhurko, Vaughan and Volec about subhypergraphs with linear minimum codegree in uniformly dense hypergraphs in the negative.