{"paper":{"title":"On Mubayi's Conjecture and conditionally intersecting sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Adam Mammoliti, Thomas Britz (UNSW Sydney)","submitted_at":"2017-11-15T08:20:20Z","abstract_excerpt":"Mubayi's Conjecture states that if $\\mathcal{F}$ is a family of $k$-sized subsets of $[n] = \\{1,\\ldots,n\\}$ which, for $k \\geq d \\geq 2$, satisfies $A_1 \\cap\\cdots\\cap A_d \\neq \\emptyset$ whenever $|A_1 \\cup\\cdots\\cup A_d| \\leq 2k$ for all distinct sets $A_1,\\ldots,A_d \\in\\mathcal{F}$, then $|\\mathcal{F}|\\leq \\binom{n-1}{k-1}$, with equality occurring only if $\\mathcal{F}$ is the family of all $k$-sized subsets containing some fixed element. This paper proves that Mubayi's Conjecture is true for all families that are invariant with respect to shifting; indeed, these families satisfy a stronger"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.05442","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}