{"paper":{"title":"On Piercing Numbers of Families Satisfying the $(p,q)_r$ Property","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Chaya Keller, Shakhar Smorodinsky","submitted_at":"2017-03-18T19:06:33Z","abstract_excerpt":"The Hadwiger-Debrunner number $HD_d(p,q)$ is the minimal size of a piercing set that can always be guaranteed for a family of compact convex sets in $\\mathbb{R}^d$ that satisfies the $(p,q)$ property. Hadwiger and Debrunner showed that $HD_d(p,q) \\geq p-q+1$ for all $q$, and equality is attained for $q > \\frac{d-1}{d}p +1$. Almost tight upper bounds for $HD_d(p,q)$ for a `sufficiently large' $q$ were obtained recently using an enhancement of the celebrated Alon-Kleitman theorem, but no sharp upper bounds for a general $q$ are known.\n  In [L. Montejano and P. Sober\\'{o}n, Piercing numbers for b"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.06338","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"}