New lower bounds for the Turán density of PGm(q)

Abstract

Let H be an r-uniform hypergraph. The Turán number ex(n,H) is the maximum number of edges in an n-vertex H-free r-uniform hypergraph. The Turán density of H is defined byπ(H)=limn→∞⁡ex(n,H)(nr). In this paper, we consider the Turán density of projective geometries. We give two new constructions of PGm(q)-free hypergraphs which improve some results given by Keevash (2005) [14]. Based on an upper bound of blocking sets of PGm(q), we give a new general lower bound for the Turán density of PGm(q). By a detailed analysis of the structures of complete arcs in PG2(q), we also get better lower bounds for the Turán density of PG2(q) with q=3,4,5.