An improved bound for regular decompositions of $3$-uniform hypergraphs of bounded $VC_2$-dimension

TL;DR

This paper establishes a significantly improved bound for regular decompositions of 3-uniform hypergraphs with bounded VC_2-dimension, reducing the complexity from Wowzer to polynomial bounds, akin to efficient graph regularity lemmas.

Contribution

It proves a new regularity lemma for 3-uniform hypergraphs with bounded VC_2-dimension, providing polynomial bounds on the complexity of regular partitions.

Findings

Bound on $ ext{VC}_2$-dimension implies polynomial bound on regularity complexity

Improves previous Wowzer-type bounds to polynomial bounds

Extends efficient regularity lemmas to higher arity hypergraphs

Abstract

A regular partition $\mathcal{P}$ for a $3$-uniform hypergraph $H=(V,E)$ consists of a partition $V=V_1\cup \ldots \cup V_t$ and for each $ij\in {[t]\choose 2}$, a partition $K_2[V_i,V_j]=P_{ij}^1\cup \ldots \cup P_{ij}^{\ell}$, such that certain quasirandomness properties hold. The complexity of $\mathcal{P}$ is the pair $(t,\ell)$. In this paper we show that if a $3$-uniform hypergraph $H$ has $VC_2$-dimension at most $k$, then there is a regular partition $\mathcal{P}$ for $H$ of complexity $(t,\ell)$, where $\ell$ is bounded by a polynomial in the degree of regularity. This is a vast improvement on the bound arising from the proof of this regularity lemma in general, in which the bound generated for $\ell$ is of Wowzer type. This can be seen as a higher arity analogue of the efficient regularity lemmas for graphs and hypergraphs of bounded VC-dimension due to Alon-Fischer-Newman, Lov\'{a}sz-Szegedy, and Fox-Pach-Suk.