  Over the past several years, numerous authors have explored model
theoretically motivated combinatorial conditions that ensure that a graph has
an efficient regular decomposition in the sense of Szemer\'edi. In this paper
we set out a research program that explores a corresponding set of questions
for 3-uniform hypergraphs, a setting in which useful notions of regularity are
significantly more intricate.
  The main results in this paper concern certain combinatorial properties which
arose as natural higher-order generalizations of the order property in parallel
work of the authors in the arithmetic setting. Interpreted in the context of
3-uniform hypergraphs, these are tightly connected to the nature of irregular
triads. Specifically, we show that a hereditary property of 3-uniform
hypergraphs admits regular decompositions with so-called "linear error" if and
only if it does not have the functional order property.
  Along the way, we show that a hereditary property of 3-uniform hypergraphs is
homogeneous (i.e. all regular triads have density near $0$ or near $1$) if and
only it has bounded $\textrm{VC}_2$-dimension, complementing a recent
(non-quantitative) result of Chernikov and Towsner.
  We also address several questions arising from prior work on tame regularity
in hypergraphs. In particular, we characterize the hereditary properties of
$3$-uniform hypergraphs admitting the type of regular partitions appearing in
work of Fox et al. as those that have finite weak VC-dimension. This is again
analogous to a recent non-quantitative result of Chernikov and Towsner.
