{"paper":{"title":"A variant of the Hadwiger-Debrunner (p,q)-problem in the plane","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM","math.CO"],"primary_cat":"cs.CG","authors_text":"Gabriel Nivasch, Sathish Govindarajan","submitted_at":"2014-09-03T18:48:30Z","abstract_excerpt":"Let $X$ be a convex curve in the plane (say, the unit circle), and let $\\mathcal S$ be a family of planar convex bodies, such that every two of them meet at a point of $X$. Then $\\mathcal S$ has a transversal $N\\subset\\mathbb R^2$ of size at most $1.75\\cdot 10^9$.\n  Suppose instead that $\\mathcal S$ only satisfies the following \"$(p,2)$-condition\": Among every $p$ elements of $\\mathcal S$ there are two that meet at a common point of $X$. Then $\\mathcal S$ has a transversal of size $O(p^8)$. For comparison, the best known bound for the Hadwiger--Debrunner $(p, q)$-problem in the plane, with $q="},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.1194","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"}