{"paper":{"title":"Growth of regular partitions 1: Improved bounds for small slicewise VC-dimension","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.LO"],"primary_cat":"math.CO","authors_text":"C. Terry","submitted_at":"2024-04-01T17:49:50Z","abstract_excerpt":"This is Part 1 in a series of papers about sizes of regular partitions of $3$-uniform hypergraphs. Previous work of the author and Wolf, and independently Chernikov and Towsner, showed that $3$-uniform hypergraphs of small slicewise VC-dimension admit homogeneous partitions. The results of Chernikov and Towsner did not produce explicit bounds, while the work of the author and Wolf relied on a strong version of hypergraph regularity, and consequently produced a Wowzer type bound on the size of the partition. This paper gives a new proof of this result, yielding $\\e$-homogeneous partitions of si"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2404.01274","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2404.01274/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}