{"paper":{"title":"Intersection patterns of planar sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Gil Kalai, Zuzana Pat\\'akov\\'a","submitted_at":"2019-07-01T15:59:05Z","abstract_excerpt":"Let $\\mathcal A=\\{A_1,\\ldots,A_n\\}$ be a family of sets in the plane. For $0 \\leq i < n$, denote by $f_i$ the number of subsets $\\sigma$ of $\\{1,\\ldots,n\\}$ of cardinality $i+1$ that satisfy $\\bigcap_{i \\in \\sigma} A_i \\neq \\emptyset$. Let $k \\geq 2$ be an integer. We prove that if each $k$-wise and $(k+1)$-wise intersection of sets from $\\mathcal A$ is empty, or a single point, or both open and path-connected, then $f_{k+1}=0$ implies $f_k \\leq cf_{k-1}$ for some positive constant $c$ depending only on $k$. Similarly, let $b \\geq 2, k > 2b$ be integers. We prove that if each $k$-wise or $(k+1"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1907.00885","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/1907.00885/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"}