{"paper":{"title":"Growth of regular partitions 4: strong regularity and the pairs partition","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.LO"],"primary_cat":"math.CO","authors_text":"C. Terry","submitted_at":"2024-04-02T15:17:56Z","abstract_excerpt":"This paper studies bounds in a strong form of regularity for $3$-uniform hypergraphs which was developed by Frankl, Gowers, Kohayakawa, Nagle, R\\\"{o}dl, Skokan, and Schacht. Regular decompositions of this type involve two structural components: a partition on the vertex set and a partition on the pairs of vertices. The regularity of such decompositions are measured by two parameters: an $\\epsilon_1>0$ and a function $\\epsilon_2:\\mathbb{N}\\rightarrow (0,1]$. To each hereditary property $\\mathcal{H}$ of $3$-uniform hypergraphs, we associate two corresponding growth functions: $T_{\\mathcal{H}}(\\e"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2404.02030","kind":"arxiv","version":3},"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/2404.02030/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"}