{"paper":{"title":"A Dense Neighborhood Lemma: Applications of Partial Concept Classes to Domination and Chromatic Number","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"cs.DM","authors_text":"Pierre Charbit, Romain Bourneuf, St\\'ephan Thomass\\'e","submitted_at":"2025-04-03T19:31:05Z","abstract_excerpt":"In its Euclidean form, the Dense Neighborhood Lemma (DNL) asserts that if $V$ is a finite set of points of $\\mathbb{R}^N$ such that for each $v \\in V$ the ball $B(v,1)$ intersects $V$ on at least $\\delta |V|$ points, then for every $\\varepsilon >0$, the points of $V$ can be covered with $f(\\delta,\\varepsilon)$ balls $B(v,1+\\varepsilon)$ with $v \\in V$. DNL also applies to other metric spaces and to abstract set systems, where elements are compared pairwise with respect to (near) disjointness. In its strongest form, DNL provides an $\\varepsilon$-clustering with size exponential in $\\varepsilon^"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2504.02992","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/2504.02992/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"}