论文标题
$ pg_ {m}（q）$的Turán密度的新下限
New lower bounds for the Turán density of $PG_{m}(q)$
论文作者
论文摘要
令$ \ mathcal {h} $为$ r $ - 统一超图。 turán数字$ \ text {ex}（n，\ mathcal {h}）$是$ n $ vertex $ \ mathcal {h} $中的最大边数 - 免费$ r $ r $ -r $ runiform-sifter-sifter-siftergraph。 $ \ Mathcal {h} $的turán密度由\ [π（\ Mathcal {h}）= \ lim_ {n \ rightarrow \ infty} \ frac {\ frac {\ frac {\ text {ex}（ex}（ex}（ex}（ex））投影几何形状的图片密度。我们给出了$ PG_ {M}（q）$的两个新结构 - 免费的超图，这些构造改善了Keevash给出的一些结果（J.Combin。TheolodeSer。A，111：289---309，2005）。基于$ pg_m（q）$的阻塞集的上限，我们为$ pg_ {m}（q）$的Turán密度提供了新的一般下限。通过对$ pg_2（q）$中完整弧的结构的详细分析，我们还获得了$ pg_2（q）$的$ q = 3，\ 4，\ 5，\ 7，\ 7，\ 8 $的turán密度的更好下限。
Let $\mathcal{H}$ be an $r$-uniform hypergraph. The Turán number $\text{ex}(n,\mathcal{H})$ is the maximum number of edges in an $n$-vertex $\mathcal{H}$-free $r$-uniform hypergraph. The Turán density of $\mathcal{H}$ is defined by \[π(\mathcal{H})=\lim_{n\rightarrow\infty}\frac{\text{ex}(n,\mathcal{H})}{\binom{n}{r}}.\] In this paper, we consider the Turán density of projective geometries. We give two new constructions of $PG_{m}(q)$-free hypergraphs which improve some results given by Keevash (J. Combin. Theory Ser. A, 111: 289--309, 2005). Based on an upper bound of blocking sets of $PG_m(q)$, we give a new general lower bound for the Turán density of $PG_{m}(q)$. By a detailed analysis of the structures of complete arcs in $PG_2(q)$, we also get better lower bounds for the Turán density of $PG_2(q)$ with $q=3,\ 4,\ 5,\ 7,\ 8$.
