# 10-30【张　韬】五教5501 吴文俊数学重点实验室组合图论系列讲座之145

题目：Hypergraphs without Berge Theta subhypergraphs

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 which 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.