Growth of regular partitions 2: Weak regularity

This is Part 2 in a series of papers about the growth of regular partitions in hereditary properties $3$-uniform hypergraphs. The focus of this paper is the notion of weak hypergraph regularity, first developed by Chung, Chung-Graham, and Haviland-Thomason. Given a hereditary property of $3$-uniform hypergraphs $\mathcal{H}$, we define a function $M_{\mathcal{H}}:(0,1)\rightarrow \mathbb{N}$ by letting $M_{\mathcal{H}}(\epsilon)$ be the smallest integer $M$ such that all sufficiently large elements of $\mathcal{H}$ admit weak regular partitions of size at most $M$. We show the asymptotic growth rate of such a function falls into one of four categories: constant, polynomial, between single and double exponentials, or tower. These results are a crucial component in Part 3 of the series, which considers vertex partitions associated to a stronger notion of hypergraph regularity.