Upper Bounds on the Tur\'{a}n 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 $q\geq2$ 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 Tur\'{a}n 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 Tur\'{a}n density was $1-\binom{q^2}{q}^{-1}$. As a consequence of our estimate, this upper bound is improved to $(q(q+1)^{-1})^{\frac{1}{q}}$.