{"paper":{"title":"Four-vertex traces of finite sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jian Wang, Peter Frankl","submitted_at":"2023-01-14T06:28:13Z","abstract_excerpt":"Let $[n]=X_1\\cup X_2\\cup X_3$ be a partition with $\\lfloor\\frac{n}{3}\\rfloor \\leq |X_i|\\leq \\lceil\\frac{n}{3}\\rceil$ and define $\\mathcal{G}=\\{G\\subset [n]\\colon |G\\cap X_i|\\leq 1, 1\\leq i\\leq 3\\}$. It is easy to check that the trace $\\mathcal{G}_{\\mid Y}:=\\{G\\cap Y\\colon G\\in \\mathcal{G}\\}$ satisfies $|\\mathcal{G}_{\\mid Y}|\\leq 12$ for all 4-sets $Y\\subset [n]$. For $n\\geq 25$ it is proven that whenever $\\mathcal{F}\\subset 2^{[n]}$ satisfies $|\\mathcal{F}|>|\\mathcal{G}|$ then $|\\mathcal{F}_{\\mid C}|\\geq 13$ for some $C\\subset [n]$, $|C|=4$. Several further results of a similar flavor are esta"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2301.05830","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/2301.05830/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"}