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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 "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1512.04026","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-12-13T09:46:53Z","cross_cats_sorted":[],"title_canon_sha256":"7e435d3e12a34c7189eaf0d8f62edcc9d61cab44c28bd75162662fed4c4ece2d","abstract_canon_sha256":"2731509c41367d8cc12f633fe7367a92d2196ed5e7bff77620316425d13aba8c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:56:04.558431Z","signature_b64":"Vq2wUOlXiZ1SlWYutEp8PVdkgAso01o72/lalSrIbftX8CTLdN3PkLiGbbuJDBVMhVXgu2xBokoTHsV/QudwCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"aa4e1027be5f8c3187b262f95932fa7096db081ae3a911f50ef12981721dfb6c","last_reissued_at":"2026-05-18T00:56:04.557865Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:56:04.557865Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1512.04026","created_at":"2026-05-18T00:56:04.557955+00:00"},{"alias_kind":"arxiv_version","alias_value":"1512.04026v3","created_at":"2026-05-18T00:56:04.557955+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1512.04026","created_at":"2026-05-18T00:56:04.557955+00:00"},{"alias_kind":"pith_short_12","alias_value":"VJHBAJ56L6GD","created_at":"2026-05-18T12:29:44.643036+00:00"},{"alias_kind":"pith_short_16","alias_value":"VJHBAJ56L6GDDB5S","created_at":"2026-05-18T12:29:44.643036+00:00"},{"alias_kind":"pith_short_8","alias_value":"VJHBAJ56","created_at":"2026-05-18T12:29:44.643036+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/VJHBAJ56L6GDDB5SML4VSMX2OC","json":"https://pith.science/pith/VJHBAJ56L6GDDB5SML4VSMX2OC.json","graph_json":"https://pith.science/api/pith-number/VJHBAJ56L6GDDB5SML4VSMX2OC/graph.json","events_json":"https://pith.science/api/pith-number/VJHBAJ56L6GDDB5SML4VSMX2OC/events.json","paper":"https://pith.science/paper/VJHBAJ56"},"agent_actions":{"view_html":"https://pith.science/pith/VJHBAJ56L6GDDB5SML4VSMX2OC","download_json":"https://pith.science/pith/VJHBAJ56L6GDDB5SML4VSMX2OC.json","view_paper":"https://pith.science/paper/VJHBAJ56","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1512.04026&json=true","fetch_graph":"https://pith.science/api/pith-number/VJHBAJ56L6GDDB5SML4VSMX2OC/graph.json","fetch_events":"https://pith.science/api/pith-number/VJHBAJ56L6GDDB5SML4VSMX2OC/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/VJHBAJ56L6GDDB5SML4VSMX2OC/action/timestamp_anchor","attest_storage":"https://pith.science/pith/VJHBAJ56L6GDDB5SML4VSMX2OC/action/storage_attestation","attest_author":"https://pith.science/pith/VJHBAJ56L6GDDB5SML4VSMX2OC/action/author_attestation","sign_citation":"https://pith.science/pith/VJHBAJ56L6GDDB5SML4VSMX2OC/action/citation_signature","submit_replication":"https://pith.science/pith/VJHBAJ56L6GDDB5SML4VSMX2OC/action/replication_record"}},"created_at":"2026-05-18T00:56:04.557955+00:00","updated_at":"2026-05-18T00:56:04.557955+00:00"}