{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:VJHBAJ56L6GDDB5SML4VSMX2OC","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"2731509c41367d8cc12f633fe7367a92d2196ed5e7bff77620316425d13aba8c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-12-13T09:46:53Z","title_canon_sha256":"7e435d3e12a34c7189eaf0d8f62edcc9d61cab44c28bd75162662fed4c4ece2d"},"schema_version":"1.0","source":{"id":"1512.04026","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1512.04026","created_at":"2026-05-18T00:56:04Z"},{"alias_kind":"arxiv_version","alias_value":"1512.04026v3","created_at":"2026-05-18T00:56:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1512.04026","created_at":"2026-05-18T00:56:04Z"},{"alias_kind":"pith_short_12","alias_value":"VJHBAJ56L6GD","created_at":"2026-05-18T12:29:44Z"},{"alias_kind":"pith_short_16","alias_value":"VJHBAJ56L6GDDB5S","created_at":"2026-05-18T12:29:44Z"},{"alias_kind":"pith_short_8","alias_value":"VJHBAJ56","created_at":"2026-05-18T12:29:44Z"}],"graph_snapshots":[{"event_id":"sha256:a29ea90cf5a94dc51bd5222537edecab74d1a31ad0b1fcba129bc6c8a68ff4cd","target":"graph","created_at":"2026-05-18T00:56:04Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"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 ","authors_text":"Chaya Keller, Gabor Tardos, Shakhar Smorodinsky","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-12-13T09:46:53Z","title":"Improved bounds on the Hadwiger-Debrunner numbers"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.04026","kind":"arxiv","version":3},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:0223aaa6b87240709ac0cd49e660200e321efd1797a022efbaa3d223abe86e9e","target":"record","created_at":"2026-05-18T00:56:04Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"2731509c41367d8cc12f633fe7367a92d2196ed5e7bff77620316425d13aba8c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-12-13T09:46:53Z","title_canon_sha256":"7e435d3e12a34c7189eaf0d8f62edcc9d61cab44c28bd75162662fed4c4ece2d"},"schema_version":"1.0","source":{"id":"1512.04026","kind":"arxiv","version":3}},"canonical_sha256":"aa4e1027be5f8c3187b262f95932fa7096db081ae3a911f50ef12981721dfb6c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"aa4e1027be5f8c3187b262f95932fa7096db081ae3a911f50ef12981721dfb6c","first_computed_at":"2026-05-18T00:56:04.557865Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:56:04.557865Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Vq2wUOlXiZ1SlWYutEp8PVdkgAso01o72/lalSrIbftX8CTLdN3PkLiGbbuJDBVMhVXgu2xBokoTHsV/QudwCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:56:04.558431Z","signed_message":"canonical_sha256_bytes"},"source_id":"1512.04026","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0223aaa6b87240709ac0cd49e660200e321efd1797a022efbaa3d223abe86e9e","sha256:a29ea90cf5a94dc51bd5222537edecab74d1a31ad0b1fcba129bc6c8a68ff4cd"],"state_sha256":"b231cd2cbdfd227680ba6bafe1e53cf32154ee48d21be96c0f32960546c93044"}