{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:QWBT3GTI5YSORL3UETWRKIXTHW","short_pith_number":"pith:QWBT3GTI","schema_version":"1.0","canonical_sha256":"85833d9a68ee24e8af7424ed1522f33daf20d2c727dc5446ec6ab3c404201385","source":{"kind":"arxiv","id":"1208.2920","version":3},"attestation_state":"computed","paper":{"title":"Fooling sets and rank","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CC"],"primary_cat":"math.CO","authors_text":"Aya Hamed, Dirk Oliver Theis, Mirjam Friesen, Troy Lee","submitted_at":"2012-08-14T16:49:22Z","abstract_excerpt":"An $n\\times n$ matrix $M$ is called a \\textit{fooling-set matrix of size $n$} if its diagonal entries are nonzero and $M_{k,\\ell} M_{\\ell,k} = 0$ for every $k\\ne \\ell$. Dietzfelbinger, Hromkovi{\\v{c}}, and Schnitger (1996) showed that $n \\le (\\mbox{rk} M)^2$, regardless of over which field the rank is computed, and asked whether the exponent on $\\mbox{rk} M$ can be improved.\n  We settle this question. In characteristic zero, we construct an infinite family of rational fooling-set matrices with size $n = \\binom{\\mbox{rk} M+1}{2}$. In nonzero characteristic, we construct an infinite family of ma"},"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":"1208.2920","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-08-14T16:49:22Z","cross_cats_sorted":["cs.CC"],"title_canon_sha256":"f703cf95a8a096afb6fb83b31a57925386b119079cdbb96d53e9493d39c197ae","abstract_canon_sha256":"96b2aeaaca4ace7bc4ed32ee0aac9866663ea430f2e579bc6db722168a31c3a1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:02:05.831808Z","signature_b64":"D7ihaIsrSDrh8h3q3VR/HDXTy/apwpnYoIMl9Ex1aRcfFSZKVQb4RXVirywqxRfv3AEPlMvhTvP6y9uo6/7MCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"85833d9a68ee24e8af7424ed1522f33daf20d2c727dc5446ec6ab3c404201385","last_reissued_at":"2026-05-18T03:02:05.831130Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:02:05.831130Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Fooling sets and rank","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CC"],"primary_cat":"math.CO","authors_text":"Aya Hamed, Dirk Oliver Theis, Mirjam Friesen, Troy Lee","submitted_at":"2012-08-14T16:49:22Z","abstract_excerpt":"An $n\\times n$ matrix $M$ is called a \\textit{fooling-set matrix of size $n$} if its diagonal entries are nonzero and $M_{k,\\ell} M_{\\ell,k} = 0$ for every $k\\ne \\ell$. Dietzfelbinger, Hromkovi{\\v{c}}, and Schnitger (1996) showed that $n \\le (\\mbox{rk} M)^2$, regardless of over which field the rank is computed, and asked whether the exponent on $\\mbox{rk} M$ can be improved.\n  We settle this question. In characteristic zero, we construct an infinite family of rational fooling-set matrices with size $n = \\binom{\\mbox{rk} M+1}{2}$. In nonzero characteristic, we construct an infinite family of ma"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1208.2920","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":"1208.2920","created_at":"2026-05-18T03:02:05.831231+00:00"},{"alias_kind":"arxiv_version","alias_value":"1208.2920v3","created_at":"2026-05-18T03:02:05.831231+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1208.2920","created_at":"2026-05-18T03:02:05.831231+00:00"},{"alias_kind":"pith_short_12","alias_value":"QWBT3GTI5YSO","created_at":"2026-05-18T12:27:20.899486+00:00"},{"alias_kind":"pith_short_16","alias_value":"QWBT3GTI5YSORL3U","created_at":"2026-05-18T12:27:20.899486+00:00"},{"alias_kind":"pith_short_8","alias_value":"QWBT3GTI","created_at":"2026-05-18T12:27:20.899486+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/QWBT3GTI5YSORL3UETWRKIXTHW","json":"https://pith.science/pith/QWBT3GTI5YSORL3UETWRKIXTHW.json","graph_json":"https://pith.science/api/pith-number/QWBT3GTI5YSORL3UETWRKIXTHW/graph.json","events_json":"https://pith.science/api/pith-number/QWBT3GTI5YSORL3UETWRKIXTHW/events.json","paper":"https://pith.science/paper/QWBT3GTI"},"agent_actions":{"view_html":"https://pith.science/pith/QWBT3GTI5YSORL3UETWRKIXTHW","download_json":"https://pith.science/pith/QWBT3GTI5YSORL3UETWRKIXTHW.json","view_paper":"https://pith.science/paper/QWBT3GTI","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1208.2920&json=true","fetch_graph":"https://pith.science/api/pith-number/QWBT3GTI5YSORL3UETWRKIXTHW/graph.json","fetch_events":"https://pith.science/api/pith-number/QWBT3GTI5YSORL3UETWRKIXTHW/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/QWBT3GTI5YSORL3UETWRKIXTHW/action/timestamp_anchor","attest_storage":"https://pith.science/pith/QWBT3GTI5YSORL3UETWRKIXTHW/action/storage_attestation","attest_author":"https://pith.science/pith/QWBT3GTI5YSORL3UETWRKIXTHW/action/author_attestation","sign_citation":"https://pith.science/pith/QWBT3GTI5YSORL3UETWRKIXTHW/action/citation_signature","submit_replication":"https://pith.science/pith/QWBT3GTI5YSORL3UETWRKIXTHW/action/replication_record"}},"created_at":"2026-05-18T03:02:05.831231+00:00","updated_at":"2026-05-18T03:02:05.831231+00:00"}