New lower bounds for the Tur\'{a}n density of $PG_{m}(q)$

TL;DR

This paper establishes new lower bounds for the Turán density of projective geometries by constructing hypergraphs free of these geometries and analyzing their combinatorial structures, improving previous results in the field.

Contribution

It introduces two novel constructions of hypergraphs avoiding projective geometries and provides improved general lower bounds for their Turán densities, especially for specific cases of $PG_2(q)$.

Findings

New constructions of $PG_m(q)$-free hypergraphs

Improved lower bounds for Turán density of projective geometries

Enhanced bounds for $PG_2(q)$ with specific q values

Abstract

Let $\mathcal{H}$ be an $r$-uniform hypergraph. The Tur\'{a}n number $\text{ex}(n,\mathcal{H})$ is the maximum number of edges in an $n$-vertex $\mathcal{H}$-free $r$-uniform hypergraph. The Tur\'{a}n density of $\mathcal{H}$ is defined by \[\pi(\mathcal{H})=\lim_{n\rightarrow\infty}\frac{\text{ex}(n,\mathcal{H})}{\binom{n}{r}}.\] In this paper, we consider the Tur\'{a}n 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 Tur\'{a}n 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 Tur\'{a}n density of $PG_2(q)$ with $q=3,\ 4,\ 5,\ 7,\ 8$.