{"paper":{"title":"Matchings vs hitting sets among half-spaces in low dimensional euclidean spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CG","cs.DM"],"primary_cat":"math.CO","authors_text":"Rom Pinchasi, Shay Moran","submitted_at":"2015-07-09T13:37:33Z","abstract_excerpt":"Let $\\mathcal{F}$ be any collection of linearly separable sets of a set $P$ of $n$ points either in $\\mathbb{R}^2$, or in $\\mathbb{R}^3$. We show that for every natural number $k$ either one can find $k$ pairwise disjoint sets in $\\mathcal{F}$, or there are $O(k)$ points in $P$ that together hit all sets in $\\mathcal{F}$. The proof is based on showing a similar result for families $\\mathcal{F}$ of sets separable by pseudo-discs in $\\mathbb{R}^2$. We complement these statements by showing that analogous result fails to hold for collections of linearly separable sets in $\\mathbb{R}^4$ and higher"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.02504","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"}