{"paper":{"title":"Exact results on traces of sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jie Ma, Mingyuan Rong, Mingze Li","submitted_at":"2024-06-27T03:51:58Z","abstract_excerpt":"For non-negative integers $n$, $m$, $a$ and $b$, we write $\\left( n,m \\right) \\rightarrow \\left( a,b \\right)$ if for every family $\\mathcal{F}\\subseteq 2^{[n]}$ with $|\\mathcal{F}|\\geqslant m$ there is an $a$-element set $T\\subseteq [n]$ such that $\\left| \\mathcal{F}_{\\mid T} \\right| \\geqslant b$, where $\\mathcal{F}_{\\mid T}=\\{ F \\cap T : F \\in \\mathcal{F} \\}$. A longstanding problem in extremal set theory asks to determine $m(s)=\\lim_{n\\rightarrow +\\infty}\\frac{m(n,s)}{n}$, where $m(n,s)$ denotes the maximum integer $m$ such that $\\left( n,m \\right) \\rightarrow \\left( n-1,m-s \\right)$ holds f"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2406.18870","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/2406.18870/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"}