Abstract
A regular partition $\mathcal{P}$ of 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 $V_i\times
V_j=P_{ij}^1\cup \ldots \cup P_{ij}^{\ell}$ so that certain quasirandomness properties hold. The
\emph{complexity of $\mathcal{P}$} is the pair $(t,\ell)$. In this talk, we present results which
show that if a 3-uniform hypergraph $H$ has $VC_2$-dimension at most $k$, then there is a
regular partition 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
result can be seen as a higher arty 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.