{"paper":{"title":"The $(2,2)$ and $(4,3)$ properties in families of fat sets in the plane","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Shiliang Gao, Shira Zerbib","submitted_at":"2017-11-14T20:45:19Z","abstract_excerpt":"A family of sets satisfies the $(p,q)$ property if among every $p$ members of it some $q$ intersect. Given a number $0<r\\le 1$, a set $S\\subset \\mathbb{R}^2$ is called $r$-fat if there exists a point $c\\in S$ such that $B(c,r) \\subseteq S\\subseteq B(c,1)$, where $B(c,r)\\subset \\mathbb{R}^2$ is a disk of radius $r$ with center-point $c$. We prove constant upper bounds $C=C(r)$ on the piercing numbers in families of $r$-fat sets in $\\mathbb{R}^2$ that satisfy the $(2,2)$ or the $(4,3)$ properties. This extends results by Danzer and Karasev on the piercing numbers in intersecting families of disk"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.05308","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":""},"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"}