{"paper":{"title":"Almost intersecting families","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Andrey Kupavskii, Peter Frankl","submitted_at":"2020-04-18T22:03:41Z","abstract_excerpt":"Let $n > k > 1$ be integers, $[n] = \\{1, \\ldots, n\\}$. Let $\\mathcal F$ be a family of $k$-subsets of~$[n]$. The family $\\mathcal F$ is called intersecting if $F \\cap F' \\neq \\emptyset$ for all $F, F' \\in \\mathcal F$. It is called almost intersecting if it is not intersecting but to every $F \\in \\mathcal F$ there is at most one $F'\\in \\mathcal F$ satisfying $F \\cap F' = \\emptyset$. Gerbner et al. proved that if $n \\geq 2k + 2$ then $|\\mathcal F| \\leq {n - 1\\choose k - 1}$ holds for almost intersecting families. The main result implies the considerably stronger and best possible bound $|\\mathca"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2004.08714","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/2004.08714/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"}