AI 中文总结
本文证明了射影几何的码度Turán密度下界，并确定了特定情况下该密度的精确值。
AI 中文摘要
设\(F\)为\(k -\)一致超图（简称为\(k -\)图）。余度数图兰密度\(\gamma(F)\)是所有\(\gamma\in[0,1)\)上的上确界，使得对于任意大的\(n\)，存在一个\(n\)个顶点的不含\(F\)的\(k -\)图\(H\)，其每个\((k - 1)\)顶点子集至少包含在\(\gamma n\)条边中。设\(PG_m(q)\)是有限域\(\mathbb{F}_q\)上维度为\(m\)的射影几何。本文证明对于所有\(m\)和\(q\)，\(\gamma(PG_m(q))\geq\frac{1}{p}>0\)，其中\(p\)是\(q + 1\)的最小素因子，解决了Keevash和Zhao（2007年，《组合论杂志B辑》）提出的一个开放问题。此外，当\(q\)为奇素数幂时，我们确定了\(PG_4(q)\)的确切余度数图兰密度。
英文摘要
Let $F$ be a $k$-uniform hypergraph, abbreviated as $k$-graph. The codegree Turán density $γ(F)$ is the supremum over all $γ\in [0,1)$ 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 $γn$ edges. Let $PG_m(q)$ be the projective geometry of dimension $m$ over finite field $\mathbb{F}_q$. In this paper, we prove that $γ(PG_m(q)) \ge \frac{1}{p}> 0$ 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 $PG_4(q)$ when $q$ is an odd prime power.
