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

2020/06/28 by Zhang, Tao, Ge, Gennian
#05C35 #05C65 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2006.15518

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 by π(H)=lim_n→∞\fracex(n,H)\binomnr. 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 PG₂(q), we also get better lower bounds for the Turán density of PG₂(q) with q=3, 4, 5, 7, 8.

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