{"paper":{"title":"The fractional Helly number for separable convexity spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Andreas F. Holmsen, Zuzana Pat\\'akov\\'a","submitted_at":"2024-12-02T12:34:43Z","abstract_excerpt":"A convex lattice set in $\\mathbb{Z}^d$ is the intersection of a convex set in $\\mathbb{R}^d$ and the integer lattice $\\mathbb{Z}^d$. A well-known theorem of Doignon states that the Helly number of $d$-dimensional convex lattice sets equals $2^d$, while a remarkable theorem of B\\'ar\\'any and Matou\\v{s}ek states that the fractional Helly number is only $d+1$. In this paper we generalize their result to abstract convexity spaces that are equipped with a suitable separation property. We also disprove a conjecture of B\\'ar\\'any and Kalai about an existence of fractional Helly property for a family "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.01445","kind":"arxiv","version":3},"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/2412.01445/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"}