{"paper":{"title":"$r$-cross $t$-intersecting families via necessary intersection points","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bjarne Sch\\\"ulke, Pranshu Gupta, Sim\\'on Piga, Yannick Mogge","submitted_at":"2020-10-22T17:55:55Z","abstract_excerpt":"Given integers $r\\geq 2$ and $n,t\\geq 1$ we call families $\\mathcal{F}_1,\\dots,\\mathcal{F}_r\\subseteq\\mathscr{P}([n])$ $r$-cross $t$-intersecting if for all $F_i\\in\\mathcal{F}_i$, $i\\in[r]$, we have $\\vert\\bigcap_{i\\in[r]}F_i\\vert\\geq t$. We obtain a strong generalisation of the classic Hilton-Milner theorem on cross intersecting families. In particular, we determine the maximum of $\\sum_{j\\in [r]}\\vert\\mathcal{F}_j\\vert$ for $r$-cross $t$-intersecting families in the cases when these are $k$-uniform families or arbitrary subfamilies of $\\mathscr{P}([n])$. Only some special cases of these resu"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2010.11928","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2010.11928/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"}