{"paper":{"title":"Maxmum Size of a Uniform Family with Bounded VC-dimension","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Tianchi Yang, Xingxing Yu","submitted_at":"2025-08-20T01:15:44Z","abstract_excerpt":"In 1984, Frankl and Pach proved that, for positive integers $n$ and $d$, the maximum size of a $(d+1)$-uniform set family $\\mathcal{F}$ on an $n$-element set with VC-dimension at most $d$ is at most ${n\\choose d}$; and they suspected that ${n\\choose d}$ could be replaced by ${n-1\\choose d}$, which would generalize the famous Erd\\H{o}s-Ko-Rado theorem and was mentioned by Erd\\H{o}s as Frankl--Pach conjecture. However, Ahlswede and Khachatrian in 1997 constructed $(d+1)$-uniform families on an $n$-element set with VC-dimension at most $d$ and size exactly $\\binom{n-1}{d}+\\binom{n-4}{d-2}$, and M"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2508.14334","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/2508.14334/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"}