The hypergraph regularity lemma—the extension of Szemerédi’s graph regularity lemma to the setting of k-uniform hypergraphs—is one of the most celebrated combinatorial results obtained in the past decade. By now there are several (very different) proofs of this lemma, obtained by Gowers, by Nagle–Rödl–Schacht–Skokan and by Tao. Unfortunately, what all these proofs have in common is that they yield regular partitions whose order is given by the k-th Ackermann function. We show that such Ackermann-type bounds are unavoidable for every \(k \ge 2\), thus confirming a prediction of Tao. Prior to our work, the only result of this type was Gowers’ famous lower bound for graph regularity.

中文翻译：

---

肥大的规律性的严格限制

超图正则性引理（将Szemerédi的图正则性引理扩展到k-一致超图的设置）是过去十年中获得的最著名的组合结果之一。到现在为止，由高尔斯，纳格勒-罗德尔-沙赫特-斯科坎和陶获得了这个引理的几个（非常不同的）证明。不幸的是，所有这些证明的共同点是它们产生规则的分区，其顺序由第k个Ackermann函数给出。我们证明，对于每个\（k \ ge 2 \），这样的Ackermann型边界都是不可避免的，从而证实了对Tao的预测。在我们进行工作之前，这种类型的唯一结果是高尔斯著名的图正则性下界。