On the Codegree Density of $PG_m(q)$

Source abstract

For an $r$-graph $G$, the minimum $(r-1)$-degree $\delta(G)$ is the largest integer $t$ such that every $(r-1)$-subset of $V(G)$ is contained in at least $t$ edges of $G$. Given an $r$-graph $F$, the codegree density $\gamma(F)$ is the largest $\gamma>0$ such that there are $F$-free $r$-graphs $G$ on $n$ vertices with $\delta(G)\ge(\gamma-o(1))n$. In this paper, we consider the codegree density of projective geometries. Employing the moment identity of a subset of $PG_{m}(q)$, we prove (1) $\gamma(PG_{2}(q))=\frac{1}{2}$ for prime power $q\equiv2\pmod{3}$; and (2) $\gamma(PG_{3}(q))=\frac{2}{3}$ for prime power $q\equiv1\pmod{2}$ or $q\equiv2\pmod{3}$. Our results partially solve an open problem proposed by Keevash and Zhao [ J. Combin. Theory Ser. B, 97 (2007), pp. 919--928]. Previously, the codegree density problems for projective geometries were settled only for $PG_{2}(2)$, $PG_{3}(2)$, $PG_{3}(3),$ and $PG_{2}(q)$ with odd prime power $q$.