Extremal constructions for apex partite hypergraphs
Qiyuan Chen, Hong Liu, Ke Ye
TL;DR
The paper advances extremal hypergraph theory by deriving new lower bounds for the Turán numbers $ex(n,\mathcal{H}(k))$ of apex partite hypergraphs and establishing their optimality for Sidorenko hypergraphs. It extends the random algebraic method of Bukh to hypergraphs and leverages algebro-geometric invariants to bound non-regular polynomial sequences, enabling precise exponent control $n^{d-\frac{1}{e(\mathcal{H})}}$ whenever $k$ is sufficiently large. For complete multipartite hypergraphs, the authors show $ex(n,\mathcal{K}^{(d)}_{s_1,\dots,s_d})=\Theta_{s_1,\dots,s_{d-1}}\bigl(n^{d-|1/(s_1\cdots s_{d-1})|}\bigr)$ when $s_d$ grows exponentially in the product, improving previous factorial thresholds and answering Mubayi’s question; this recovers the $d=2$ case of Bukh. The work also provides a hypergraph Zarankiewicz bound in a generalized, sided setting, under explicit growth conditions on the parameters, thereby broadening the applicability of the algebraic-geometry framework to extremal problems.
Abstract
We establish new lower bounds for the Turán and Zarankiewicz numbers of certain apex partite hypergraphs. Given a $(d-1)$-partite $(d-1)$-uniform hypergraph $\mathcal{H}$, let $\mathcal{H}(k)$ be the $d$-partite $d$-uniform hypergraph whose $d$th part has $k$ vertices that share $\mathcal{ H}$ as a common link. We show that $ex(n,\mathcal{H}(k))=Ω_{\mathcal{ H}}(n^{d-\frac{1}{e(\mathcal{H})}})$ if $k$ is at least exponentially large in $e(\mathcal{H})$. Our bound is optimal for all Sidorenko hypergraphs $\mathcal{H}$ and verifies a conjecture of Lee for such hypergraphs. In particular, for the complete $d$-partite $d$-uniform hypergraphs $\mathcal{K}^{(d)}_{s_1,\dots,s_d}$, our result implies that $ex(n,\mathcal{K}^{(d)}_{s_{1},\cdots,s_{d}})=Θ(n^{d-\frac{1}{s_{1}\cdots s_{d-1}}})$ if $s_{d}$ is at least exponentially large in terms of $s_{1}\cdots s_{d-1}$, improving the factorial condition of Pohoata and Zakharov and answering a question of Mubayi. Our method is a generalization of Bukh's random algebraic method [Duke Math.J. 2024] to hypergraphs, and extends to the sided Zarankiewicz problem.