{"paper":{"title":"A new lower bound on Hadwiger-Debrunner numbers in the plane","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CG"],"primary_cat":"math.CO","authors_text":"Chaya Keller, Shakhar Smorodinsky","submitted_at":"2018-09-17T21:25:59Z","abstract_excerpt":"A family of sets $F$ is said to satisfy the $(p,q)$ property if among any $p$ sets in $F$, some $q$ have a non-empty intersection. Hadwiger and Debrunner (1957) conjectured that for any $p \\geq q \\geq d+1$ there exists $c=c_d(p,q)$, such that any family of compact convex sets in $\\mathbb{R}^d$ that satisfies the $(p,q)$ property, can be pierced by at most $c$ points. In a celebrated result from 1992, Alon and Kleitman proved the conjecture. However, obtaining sharp bounds on $c_d(p,q)$, called `the Hadwiger-Debrunner numbers', is still a major open problem in discrete and computational geometr"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.06451","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":""},"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"}