{"paper":{"title":"Intersecting families with large shadow degree","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jian Wang, Peter Frankl","submitted_at":"2024-06-01T15:25:21Z","abstract_excerpt":"A $k$-uniform family $\\mathcal{F}$ is called intersecting if $F\\cap F'\\neq \\emptyset$ for all $F,F'\\in \\mathcal{F}$. The shadow family $\\partial \\mathcal{F}$ is the family of $(k-1)$-element sets that are contained in some members of $\\mathcal{F}$. The shadow degree (or minimum positive co-degree) of $\\mathcal{F}$ is defined as the maximum integer $r$ such that every $E\\in \\partial \\mathcal{F}$ is contained in at least $r$ members of $\\mathcal{F}$. In 2021, Balogh, Lemons and Palmer determined the maximum size of an intersecting $k$-uniform family with shadow degree at least $r$ for $n\\geq n_0"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2406.00465","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/2406.00465/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"}