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 H, we define a function M₇: (0, 1) N by letting M₇ () be the smallest integer M such that all sufficiently large elements of 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.