Uniform hypergraphs of girth $6$ and $8$ from generalized polygons

TL;DR

This paper investigates the maximum size of uniform hypergraphs with girth 6 and 8, establishing lower bounds that grow polynomially with the number of vertices, using constructions from generalized polygons.

Contribution

It provides new asymptotic lower bounds for the maximum edges in uniform hypergraphs with girth 6 and 8, based on generalized polygons, for large vertex counts.

Findings

Lower bounds for girth 6 hypergraphs: approximately N^{11/8}

Lower bounds for girth 8 hypergraphs: approximately N^{11/9}

Bounds include logarithmic correction factors

Abstract

Let $ex_r(N,g)$ be the maximum number of edges in an $r$-uni\-form hypergraph on $N$ vertices with girth at least $g$. We are interested in the asymptotic behavior of this value when $N$ is increasing but parameters $g\in\{6,8\}$ and $r\geq3$ are fixed. It is shown that for some positive constants $c$ and $d$, any integer $r\geq3$ and all sufficiently large integers $N$ the inequalities $ex_r(N,6)\geq N^{\frac{11}{8}-\frac{c}{\sqrt{\log N}}}$ and $ex_r(N,8)\geq N^{\frac{11}{9}-\frac{d}{\sqrt{\log N}}}$ hold.