Upper Bounds on the Turán Density of Hypergraphs Associated with the Projective Plane over a Finite Field

Abstract

Let $\PG(2,q)$ denote the projective plane over the finite field ${\mathbb{F}}_q$ where $q\ge 2$ is a prime power. For a positive integer $t$, let $B_{t}[\PG(2,q)]$ denote the $t$-page book obtained from $t-$many copies of the $(q+1)-$graph $\PG(2,q)$, sharing a common edge. Employing the ${\mathsf{L}}^p-$method and using the incidence structure of the projective plane, we establish that the upper bound of Turán density of $B_{t}[\PG(2,q)]$ is $(q(q+1{)}^{-1}{)}^{\frac{1}{qt}}$. For the $(q+1)-$graph $\PG(2,q)$, the previously known upper bound on its Turán density was $1-{\left(\genfrac{}{}{0ex}{}{q^2}{q}\right)}^{-1}$. As a consequence of our estimate, this upper bound is improved to $(q(q+1{)}^{-1}{)}^{\frac{1}{q}}$.

arXiv commentKey words: r-graph, Turán density, Projective Plane over a Finite Field