Regularity for hypergraphs with bounded VC$_2$ dimension

Abstract

While Szemer\'edi's graph regularity lemma is an indispensable tool for studying extremal problems in graph theory, using it comes with a hefty price, since a worst-case graph may only have regular partitions of tower-type size. It is thus sensible to ask if there is some natural restriction which forces graphs to have much smaller regular partitions. A celebrated result of this type, due to Alon-Fischer-Newman and Lov\'asz-Szegedy, states that for graphs of bounded VC dimension, one can reduce the tower-type bounds to polynomial. The graph regularity lemma has been extended to the setting of $k$-graphs by Gowers, Nagle-R\"odl-Schacht-Skokan, and Tao. Unfortunately, these lemmas come with even larger Ackermann-type bounds. Chernikov-Starchenko and Fox-Pach-Suk considered a strong notion of $k$-graph VC dimension and proved that $k$-graphs of bounded VC dimension have regular partitions of polynomial size. Shelah introduced a weaker and combinatorially natural notion of dimension, called VC$_2$ dimension, which has since been extensively studied. In particular, Chernikov, Towsner, Terry, and Wolf asked if one can improve the worst case bounds for 3-graph regularity when the 3-graph has bounded VC$_2$ dimension. Our main result in this paper answers this question positively in the following strong sense: in the setting of bounded VC$_2$ dimension, one can reduce the bounds for 3-graph regularity by one level in Ackermann hierarchy. Furthermore, our new bound is best possible. Our proof has two key steps. We first introduce a new method for designing regularity lemmas for graphs of bounded VC dimension, based on the cylinder regularity lemma. We then prove a hypergraph version of the cylinder regularity lemma, which allows us to extend this method to hypergraphs. We also highlight a few other applications of this cylinder regularity lemma, which we expect to find many other uses.