Norm hypergraphs

We introduce a high uniformity generalization of the so-called (projective) norm graphs of Alon, Kollár, Rónyai, and Szabó, and use it to show that $\operatorname{ex}_{d}(n,K_{s_{1},\ldots,s_{d}}^{(d)}) = Θ\left(n^{d - \frac{1}{s_{1}\ldots s_{d-1}}}\right)$ 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.

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