Regularity for hypergraphs with bounded VC$_2$ dimension
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| Format: | Preprint |
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2025
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| author | Gishboliner, Lior Shapira, Asaf Wigderson, Yuval |
| author_facet | Gishboliner, Lior Shapira, Asaf Wigderson, Yuval |
| contents | While Szemerédi'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 Lovász-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-Rödl-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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_09969 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Regularity for hypergraphs with bounded VC$_2$ dimension Gishboliner, Lior Shapira, Asaf Wigderson, Yuval Combinatorics While Szemerédi'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 Lovász-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-Rödl-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. |
| title | Regularity for hypergraphs with bounded VC$_2$ dimension |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2508.09969 |