3-uniform hypergraphs with few Berge paths of length three between any two vertices

TL;DR

This paper determines the Turán number for 3-uniform Berge theta hypergraphs with specific parameters, providing explicit algebraic constructions that match known upper bounds and introducing the use of polynomial resultants as a key technique.

Contribution

It offers the first explicit algebraic construction for the Turán number of certain Berge theta hypergraphs, improving understanding of their extremal properties.

Findings

Established a lower bound of (n^{4/3}) for the Ture1n number of -uniform Berge theta hypergraphs.

Provided an algebraic construction based on finite field equations that matches upper bounds up to a constant.

Introduced the use of polynomial resultants as a novel technique in hypergraph extremal problems.

Abstract

Recently, Berge theta hypergraphs have received special attention due to the similarity with Berge even cycles. Let $r$-uniform Berge theta hypergraph $\Theta_{\ell,t}^{B}$ be the $r$-uniform hypergraph consisting of $t$ internally disjoint Berge paths of length $\ell$ with the same pair of endpoints. In this work, we determine the Tur\'{a}n number of $3$-uniform Berge theta hypergraph when $\ell=3$ and $t$ is relatively small. More precisely, we provide an explicit construction giving \begin{align*} \textup{ex}_{3}(n,\Theta_{3,217}^{B})=\Omega(n^{\frac{4}{3}}). \end{align*} This matches an earlier upper bound by He and Tait up to an absolute constant factor. The construction is algebraic, which is based on some equations over finite fields, and the parameter $t$ in our construction is much smaller than that in random algebraic construction. Our main technique is using the resultant of polynomials, which appears to be a powerful technique to eliminate variables.