{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:3ENA37EVNLQEIA6QJOAAIKKBYT","short_pith_number":"pith:3ENA37EV","canonical_record":{"source":{"id":"1604.01257","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-04-05T13:48:11Z","cross_cats_sorted":[],"title_canon_sha256":"e5f642d41478b8c9d668f1edac480a171ad5f9dff73812a93007f838e09dc02b","abstract_canon_sha256":"ff79ccefd41f03426955babfb2bd8e61e54af60a78a74e9493bbec2037b56247"},"schema_version":"1.0"},"canonical_sha256":"d91a0dfc956ae04403d04b80042941c4ce1a140db73ae8cfbbf3afa66d6e8ddd","source":{"kind":"arxiv","id":"1604.01257","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1604.01257","created_at":"2026-05-18T01:17:38Z"},{"alias_kind":"arxiv_version","alias_value":"1604.01257v1","created_at":"2026-05-18T01:17:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1604.01257","created_at":"2026-05-18T01:17:38Z"},{"alias_kind":"pith_short_12","alias_value":"3ENA37EVNLQE","created_at":"2026-05-18T12:29:55Z"},{"alias_kind":"pith_short_16","alias_value":"3ENA37EVNLQEIA6Q","created_at":"2026-05-18T12:29:55Z"},{"alias_kind":"pith_short_8","alias_value":"3ENA37EV","created_at":"2026-05-18T12:29:55Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:3ENA37EVNLQEIA6QJOAAIKKBYT","target":"record","payload":{"canonical_record":{"source":{"id":"1604.01257","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-04-05T13:48:11Z","cross_cats_sorted":[],"title_canon_sha256":"e5f642d41478b8c9d668f1edac480a171ad5f9dff73812a93007f838e09dc02b","abstract_canon_sha256":"ff79ccefd41f03426955babfb2bd8e61e54af60a78a74e9493bbec2037b56247"},"schema_version":"1.0"},"canonical_sha256":"d91a0dfc956ae04403d04b80042941c4ce1a140db73ae8cfbbf3afa66d6e8ddd","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:17:38.718060Z","signature_b64":"fWxLHWnW9DxzgXYWkopq73bmNlfXjo+ZCi2eSP6S2pT28lHFH5rnykBwhGyETWKkt00bOlEpx9iynWoJG9LjBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d91a0dfc956ae04403d04b80042941c4ce1a140db73ae8cfbbf3afa66d6e8ddd","last_reissued_at":"2026-05-18T01:17:38.717278Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:17:38.717278Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1604.01257","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:17:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"e8Tq7qTlTlAGyLT+tBh0I5r8fncJnocaEC9ZPG3HSsl/nAfwSMmThsaBSwYmIIuyLHI0QYpxVupGuct1nnwqCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T07:50:07.824630Z"},"content_sha256":"c415a14301e10ea1a1cde13ec8d4c11947152065dff2d1860fa1ddc8c269911a","schema_version":"1.0","event_id":"sha256:c415a14301e10ea1a1cde13ec8d4c11947152065dff2d1860fa1ddc8c269911a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:3ENA37EVNLQEIA6QJOAAIKKBYT","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Zarankiewicz Numbers and Bipartite Ramsey Numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Alexander Riasanovsky, Alex Collins, John Wallace, Stanis{\\l}aw Radziszowski","submitted_at":"2016-04-05T13:48:11Z","abstract_excerpt":"The Zarankiewicz number $z(b;s)$ is the maximum size of a subgraph of $K_{b,b}$ which does not contain $K_{s,s}$ as a subgraph. The two-color bipartite Ramsey number $b(s,t)$ is the smallest integer $b$ such that any coloring of the edges of $K_{b,b}$ with two colors contains a $K_{s,s}$ in the first color or a $K_{t,t}$ in the second color.\n  In this work, we design and exploit a computational method for bounding and computing Zarankiewicz numbers. Using it, we obtain several new values and bounds on $z(b;s)$ for $3 \\le s \\le 6$. Our approach and new knowledge about $z(b;s)$ permit us to impr"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.01257","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:17:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"iNyyGTbnoFBqFZEFp1W5SUAj/qYWs28fTVi6P394dmUc9moVmYufnAbx9Jiu5PihQ9U1nI65bEmAKdnOazgIBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T07:50:07.825345Z"},"content_sha256":"52cbcf5f1834c538e400248505f8049c8b333d38b72e142170d3fe30373d1249","schema_version":"1.0","event_id":"sha256:52cbcf5f1834c538e400248505f8049c8b333d38b72e142170d3fe30373d1249"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/3ENA37EVNLQEIA6QJOAAIKKBYT/bundle.json","state_url":"https://pith.science/pith/3ENA37EVNLQEIA6QJOAAIKKBYT/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/3ENA37EVNLQEIA6QJOAAIKKBYT/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-09T07:50:07Z","links":{"resolver":"https://pith.science/pith/3ENA37EVNLQEIA6QJOAAIKKBYT","bundle":"https://pith.science/pith/3ENA37EVNLQEIA6QJOAAIKKBYT/bundle.json","state":"https://pith.science/pith/3ENA37EVNLQEIA6QJOAAIKKBYT/state.json","well_known_bundle":"https://pith.science/.well-known/pith/3ENA37EVNLQEIA6QJOAAIKKBYT/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:3ENA37EVNLQEIA6QJOAAIKKBYT","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":"ff79ccefd41f03426955babfb2bd8e61e54af60a78a74e9493bbec2037b56247","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-04-05T13:48:11Z","title_canon_sha256":"e5f642d41478b8c9d668f1edac480a171ad5f9dff73812a93007f838e09dc02b"},"schema_version":"1.0","source":{"id":"1604.01257","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1604.01257","created_at":"2026-05-18T01:17:38Z"},{"alias_kind":"arxiv_version","alias_value":"1604.01257v1","created_at":"2026-05-18T01:17:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1604.01257","created_at":"2026-05-18T01:17:38Z"},{"alias_kind":"pith_short_12","alias_value":"3ENA37EVNLQE","created_at":"2026-05-18T12:29:55Z"},{"alias_kind":"pith_short_16","alias_value":"3ENA37EVNLQEIA6Q","created_at":"2026-05-18T12:29:55Z"},{"alias_kind":"pith_short_8","alias_value":"3ENA37EV","created_at":"2026-05-18T12:29:55Z"}],"graph_snapshots":[{"event_id":"sha256:52cbcf5f1834c538e400248505f8049c8b333d38b72e142170d3fe30373d1249","target":"graph","created_at":"2026-05-18T01:17:38Z","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":"The Zarankiewicz number $z(b;s)$ is the maximum size of a subgraph of $K_{b,b}$ which does not contain $K_{s,s}$ as a subgraph. The two-color bipartite Ramsey number $b(s,t)$ is the smallest integer $b$ such that any coloring of the edges of $K_{b,b}$ with two colors contains a $K_{s,s}$ in the first color or a $K_{t,t}$ in the second color.\n  In this work, we design and exploit a computational method for bounding and computing Zarankiewicz numbers. Using it, we obtain several new values and bounds on $z(b;s)$ for $3 \\le s \\le 6$. Our approach and new knowledge about $z(b;s)$ permit us to impr","authors_text":"Alexander Riasanovsky, Alex Collins, John Wallace, Stanis{\\l}aw Radziszowski","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-04-05T13:48:11Z","title":"Zarankiewicz Numbers and Bipartite Ramsey Numbers"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.01257","kind":"arxiv","version":1},"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:c415a14301e10ea1a1cde13ec8d4c11947152065dff2d1860fa1ddc8c269911a","target":"record","created_at":"2026-05-18T01:17:38Z","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":"ff79ccefd41f03426955babfb2bd8e61e54af60a78a74e9493bbec2037b56247","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-04-05T13:48:11Z","title_canon_sha256":"e5f642d41478b8c9d668f1edac480a171ad5f9dff73812a93007f838e09dc02b"},"schema_version":"1.0","source":{"id":"1604.01257","kind":"arxiv","version":1}},"canonical_sha256":"d91a0dfc956ae04403d04b80042941c4ce1a140db73ae8cfbbf3afa66d6e8ddd","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d91a0dfc956ae04403d04b80042941c4ce1a140db73ae8cfbbf3afa66d6e8ddd","first_computed_at":"2026-05-18T01:17:38.717278Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:17:38.717278Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"fWxLHWnW9DxzgXYWkopq73bmNlfXjo+ZCi2eSP6S2pT28lHFH5rnykBwhGyETWKkt00bOlEpx9iynWoJG9LjBA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:17:38.718060Z","signed_message":"canonical_sha256_bytes"},"source_id":"1604.01257","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c415a14301e10ea1a1cde13ec8d4c11947152065dff2d1860fa1ddc8c269911a","sha256:52cbcf5f1834c538e400248505f8049c8b333d38b72e142170d3fe30373d1249"],"state_sha256":"0fcf722986636b71240b5d704d21ac4e1028f2cb4b7e28d9e1c185fb0521b571"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"N3GwFQ2gWVZJSuLmgA02LcQZl+ctj+s15S+6DUafZZLHZ0KyiYcIW4zCkIh0TRLMajrt+dizDt0xK218wI1GBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-09T07:50:07.828888Z","bundle_sha256":"217f9fa605ba3db5aeab1e074f325c1877fde18150c0eeb938914e39e5733a8d"}}