New lower bounds for the Turán density of $P{G}_m(q)$

Abstract

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 $P{G}_m(q)$-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 $P{G}_m(q)$, we give a new general lower bound for the Turán density of $P{G}_m(q)$. By a detailed analysis of the structures of complete arcs in $P{G}_2(q)$, we also get better lower bounds for the Turán density of $P{G}_2(q)$ with $q=3,4,5,7,8$.

15 pages