Year: 2026
Type: preprint
Abstract: For a prime power $q\geq2$ and an integer $m\geq2$, let $PG(m,q)$ denote the $m-$dimensional projective hypergraph over the finite field $\mathbb{F}_q$. Employing the $\mathsf{L}^p-$method and using the incidence structure, we establish the Turán density of $PG(m,q)$ \begin{equation*} π(PG(m,q))\leq\begin{cases} \left(1-\frac{1}{q+1}\right)^{\frac{1}{q}} & \text{ if } m=2, \left(1-\frac{1}{(q+1)|E(PG(m-1,q))|}\right)^{\frac{1}{1+q^{m-1}}} & \text{ if } m\geq3. \end{cases} \end{equation*} The previously known upper bound on its Turán density was $1-\binom{q^m}{q}^{-1}$. As a consequence of our estimate, this upper bound is improved for each prime power $q\geq3$ and for each dimension index $m\geq2$. For a positive integer $t$, let $B_{t}[PG(2,q)]$ be the $(q+1)-$graph formed by taking $t$ copies of the $PG(2,q)$ that shares a common edge. Using the $\mathsf{L}^p-$method, we establish that the upper bound of the Turán density of $B_{t}[PG(2,q)]$ is $\left(1-\frac{1}{q+1}\right)^{\frac{1}{qt}}$.
Source: arXiv (Cornell University)
Authors: Dash, Subhankar, Majumder, Kaushik
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7215752385