Growth of regular partitions 2: Weak regularity
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866909156242358272 |
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| author | Terry, C. |
| author_facet | Terry, C. |
| contents | 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 $M_{\mathcal{H}}(ε)$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_01293 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Growth of regular partitions 2: Weak regularity Terry, C. Combinatorics Logic 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 $M_{\mathcal{H}}(ε)$ 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. |
| title | Growth of regular partitions 2: Weak regularity |
| topic | Combinatorics Logic |
| url | https://arxiv.org/abs/2404.01293 |