Norm hypergraphs

TL;DR

This paper generalizes norm graphs to high uniformity hypergraphs and determines their extremal number for certain complete d-partite hypergraphs, advancing understanding in extremal combinatorics.

Contribution

It introduces a high uniformity generalization of norm graphs and establishes new extremal bounds for complete multipartite hypergraphs, settling many cases of Mubayi's conjecture.

Findings

Established asymptotic extremal number for specific hypergraphs.

Generalized norm graphs to higher uniformities.

Improved bounds over previous results.

Abstract

We introduce a high uniformity generalization of the so-called (projective) norm graphs of Alon, Koll\'ar, R\'onyai, and Szab\'o, and use it to show that $$\operatorname{ex}_{d}(n,K_{s_{1},\ldots,s_{d}}^{(d)}) = \Theta\left(n^{d - \frac{1}{s_{1}\ldots s_{d-1}}}\right)$$ holds for all integers $s_{1},\ldots,s_{d} \geq 2$ such that $s_{d} \geq \left((d-1)(s_{1}\ldots s_{d-1}-1)\right)!+1$. This improves upon a recent result of Ma, Yuan and Zhang, and thus settles (many) new cases of a conjecture of Mubayi.