SIAM Journal on Discrete Mathematics, Volume 35, Issue 3, Page 1548-1556, January 2021.
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 $q$.

中文翻译：

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关于$PG_m(q)$的Codegree Density

SIAM 离散数学杂志，第 35 卷，第 3 期，第 1548-1556 页，2021 年 1 月。
对于$r$-graph $G$，最小$(r-1)$-degree $\delta(G)$ 是最大的整数$t$，使得$的每个$(r-1)$-subset V(G)$ 包含在 $G$ 的至少 $t$ 个边中。给定$r$-graph $F$，codegree 密度$\gamma(F)$ 是最大的$\gamma>0$，使得$n 上有$F$-free $r$-graphs $G$ $ 顶点与 $\delta(G)\ge(\gamma-o(1))n$。在本文中，我们考虑了射影几何的代码密度。利用 $PG_{m}(q)$ 的子集的矩恒等式，我们证明了 (1) $\gamma(PG_{2}(q))=\frac{1}{2}$ 的素数幂 $q \equiv2\pmod{3}$; 和 (2) $\gamma(PG_{3}(q))=\frac{2}{3}$ 为素幂 $q\equiv1\pmod{2}$ 或 $q\equiv2\pmod{3}$ . 我们的结果部分解决了 Keevash 和 Zhao 提出的一个开放问题 [J. 结合。理论系列 B, 97 (2007)，第 919--928 页]。之前，