{"paper":{"title":"Improved bounds on the Hadwiger-Debrunner numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Chaya Keller, Gabor Tardos, Shakhar Smorodinsky","submitted_at":"2015-12-13T09:46:53Z","abstract_excerpt":"Let $HD_d(p,q)$ denote the minimal size of a transversal that can always be guaranteed for a family of compact convex sets in $\\mathbb{R}^d$ which satisfy the $(p,q)$-property ($p \\geq q \\geq d+1$). In a celebrated proof of the Hadwiger-Debrunner conjecture, Alon and Kleitman proved that $HD_d(p,q)$ exists for all $p \\geq q \\geq d+1$. Specifically, they prove that $HD_d(p,d+1)$ is $\\tilde{O}(p^{d^2+d})$.\n  We present several improved bounds:\n  (i) For any $q \\geq d+1$, $HD_d(p,q) = \\tilde{O}(p^{d \\left(\\frac{q-1}{q-d}\\right)})$.\n  (ii) For $q \\geq \\log p$, $HD_d(p,q) = \\tilde{O}(p+(p/q)^d)$.\n "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.04026","kind":"arxiv","version":3},"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"}