{"paper":{"title":"Extensions of discrete Helly theorems for boxes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Pablo Sober\\'on, Timothy Edwards","submitted_at":"2024-04-22T16:10:12Z","abstract_excerpt":"We prove extensions of Halman's discrete Helly theorem for axis-parallel boxes in $\\mathbb{R}^d$. Halman's theorem says that, given a set $S$ in $\\mathbb{R}^d$, if $F$ is a finite family of axis-parallel boxes such that the intersection of any $2d$ contains a point of $S$, then the intersection of $F$ contains a point of $S$. We prove colorful, fractional, and quantitative versions of Halman's theorem. For the fractional versions, it is enough to check that many $(d+1)$-tuples of the family contain points of $S$. Among the colorful versions we include variants where the coloring condition is r"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2404.14308","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/2404.14308/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"}