{"paper":{"title":"Sharp results concerning disjoint cross-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":"2019-05-20T14:02:08Z","abstract_excerpt":"For an $n$-element set $X$ let $\\binom{X}{k}$ be the collection of all its $k$-subsets. Two families of sets $\\mathcal A$ and $\\mathcal B$ are called cross-intersecting if $A\\cap B \\neq \\emptyset$ holds for all $A\\in\\mathcal A$, $B\\in\\mathcal B$. Let $f(n,k)$ denote the maximum of $\\min\\{|\\mathcal A|, |\\mathcal B|\\}$ where the maximum is taken over all pairs of {\\em disjoint}, cross-intersecting families $\\mathcal A, \\mathcal B\\subset\\binom{[n]}{k}$. Let $c=\\log_2e$. We prove that $f(n,k)=\\left\\lfloor\\frac12\\binom{n-1}{k-1}\\right\\rfloor$ essentially iff $n>ck^2$ (cf. Theorem~1.4 for the exact "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1905.08123","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":""},"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"}