Submission date: 1st April 2024

Abstract:

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 M_ℋ: (0,1) ↠ ℕ by letting M_{ℋ}(ε) 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.

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Full text arXiv 2404.01293: pdf, ps.