On the codegree Turán density of projective geometries

Abstract

Let $F$ be a $k$-uniform hypergraph, abbreviated as $k$-graph. The codegree Turán density $\gamma (F)$ is the supremum over all $\gamma \in [0,1)$ such that, for arbitrarily large $n$, there exists an $n$-vertex $F$-free $k$-graph $H$ whose every $(k-1)$-subset of vertices lies in at least $\gamma n$ edges. Let $P{G}_m(q)$ be the projective geometry of dimension $m$ over finite field ${\mathbb{F}}_q$. In this paper, we prove that $\gamma (P{G}_m(q))\ge \frac{1}{p}>0$ for all $m$ and $q$, where $p$ is the smallest prime divisor of $q+1$. This resolves an open problem proposed by Keevash and Zhao (JCT-B, 2007). Moreover, we determine the exact codegree Turán density of $P{G}_4(q)$ when $q$ is an odd prime power.