{"paper":{"title":"Non-trivial $d$-wise Intersecting families","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jacques Verstraete, Jason O'Neill","submitted_at":"2019-11-04T05:04:21Z","abstract_excerpt":"For an integer $d \\geq 2$, a family $\\mathcal{F}$ of sets is $\\textit{$d$-wise intersecting}$ if for any distinct sets $A_1,A_2,\\dots,A_d \\in \\mathcal{F}$, $A_1 \\cap A_2 \\cap \\dots \\cap A_d \\neq \\emptyset$, and $\\textit{non-trivial}$ if $\\bigcap \\mathcal{F} = \\emptyset$. Hilton and Milner conjectured that for $k \\geq d \\geq 2$ and large enough $n$, the extremal non-trivial $d$-wise intersecting family of $k$-element subsets of $[n]$ is one of the following two families: \\begin{align*} &\\mathcal{H}(k,d) = \\{A \\in \\binom{[n]}{k} : [d-1] \\subset A, A \\cap [d,k+1] \\neq \\emptyset\\} \\cup \\{[k+1] \\se"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1911.01031","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/1911.01031/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"}