{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2022:KD3CSBJKVC4TN4BTRCK47ETT7H","short_pith_number":"pith:KD3CSBJK","canonical_record":{"source":{"id":"2202.05507","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2022-02-11T08:43:53Z","cross_cats_sorted":[],"title_canon_sha256":"8c4cd546d1310ab5fcaac38cb100fffc71a566e3f838bcb11a539990ed89e8b5","abstract_canon_sha256":"2ea27561d3542c3060716d299e3f64f73d2d1965c4c8ae3db7f571339cbc9530"},"schema_version":"1.0"},"canonical_sha256":"50f629052aa8b936f0338895cf9273f9f10c7e4262f805b94b2425e9848332ac","source":{"kind":"arxiv","id":"2202.05507","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2202.05507","created_at":"2026-07-05T08:06:10Z"},{"alias_kind":"arxiv_version","alias_value":"2202.05507v2","created_at":"2026-07-05T08:06:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2202.05507","created_at":"2026-07-05T08:06:10Z"},{"alias_kind":"pith_short_12","alias_value":"KD3CSBJKVC4T","created_at":"2026-07-05T08:06:10Z"},{"alias_kind":"pith_short_16","alias_value":"KD3CSBJKVC4TN4BT","created_at":"2026-07-05T08:06:10Z"},{"alias_kind":"pith_short_8","alias_value":"KD3CSBJK","created_at":"2026-07-05T08:06:10Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2022:KD3CSBJKVC4TN4BTRCK47ETT7H","target":"record","payload":{"canonical_record":{"source":{"id":"2202.05507","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2022-02-11T08:43:53Z","cross_cats_sorted":[],"title_canon_sha256":"8c4cd546d1310ab5fcaac38cb100fffc71a566e3f838bcb11a539990ed89e8b5","abstract_canon_sha256":"2ea27561d3542c3060716d299e3f64f73d2d1965c4c8ae3db7f571339cbc9530"},"schema_version":"1.0"},"canonical_sha256":"50f629052aa8b936f0338895cf9273f9f10c7e4262f805b94b2425e9848332ac","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T08:06:10.248855Z","signature_b64":"Wz7UPHf0jD9xYL5oN3lQW2WyidPh+/0d2oYNBaY8o7p0H9DqbxPSsKdlRsa7hQCR74qRe7u15+Eqxxf2qsSJDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"50f629052aa8b936f0338895cf9273f9f10c7e4262f805b94b2425e9848332ac","last_reissued_at":"2026-07-05T08:06:10.248453Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T08:06:10.248453Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2202.05507","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T08:06:10Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"aW6rzA2UkkySFamdrCRqQ6/V0KJSRlFzn+lWzj+offwlrQVOF3q7ZD3yw3yUYbQNR9QW8D7S3IoCMfPSbUeKCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-11T18:25:23.581303Z"},"content_sha256":"980417d6019a6f27e890a2a80787dcc2eb5b49555068483ceec6a043132a95ef","schema_version":"1.0","event_id":"sha256:980417d6019a6f27e890a2a80787dcc2eb5b49555068483ceec6a043132a95ef"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2022:KD3CSBJKVC4TN4BTRCK47ETT7H","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Exact values for unbalanced Zarankiewicz numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Adam Mammoliti, Daniel Horsley, Guangzhou Chen","submitted_at":"2022-02-11T08:43:53Z","abstract_excerpt":"For positive integers $s$, $t$, $m$ and $n$, the Zarankiewicz number $Z_{s,t}(m,n)$ is defined to be the maximum number of edges in a bipartite graph with parts of sizes $m$ and $n$ that has no complete biparitite subgraph containing $s$ vertices in the part of size $m$ and $t$ vertices in the part of size $n$. A simple argument shows that, for each $t \\geq 2$, $Z_{2,t}(m,n)=(t-1)\\binom{m}{2}+n$ when $n \\geq (t-1)\\binom{m}{2}$. Here, for large $m$, we determine the exact value of $Z_{2,t}(m,n)$ in almost all of the remaining cases where $n=\\Theta(tm^2)$. We establish a new family of upper boun"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2202.05507","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2202.05507/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T08:06:10Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jRCTtofmqqrTk3SDyD+Oj3ZBXLz4QDtWBS45fP397+iiOEPbrR3UqrMZg+xGeRV4jYT+cXiLJxR84SDWTAyPDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-11T18:25:23.581829Z"},"content_sha256":"94cd3c1f9d17f2ca7b6b534fe211bdd2e167a516b100c4bbdc9eaf943038c3db","schema_version":"1.0","event_id":"sha256:94cd3c1f9d17f2ca7b6b534fe211bdd2e167a516b100c4bbdc9eaf943038c3db"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/KD3CSBJKVC4TN4BTRCK47ETT7H/bundle.json","state_url":"https://pith.science/pith/KD3CSBJKVC4TN4BTRCK47ETT7H/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/KD3CSBJKVC4TN4BTRCK47ETT7H/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":"2026-08-11T18:25:23Z","links":{"resolver":"https://pith.science/pith/KD3CSBJKVC4TN4BTRCK47ETT7H","bundle":"https://pith.science/pith/KD3CSBJKVC4TN4BTRCK47ETT7H/bundle.json","state":"https://pith.science/pith/KD3CSBJKVC4TN4BTRCK47ETT7H/state.json","well_known_bundle":"https://pith.science/.well-known/pith/KD3CSBJKVC4TN4BTRCK47ETT7H/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2022:KD3CSBJKVC4TN4BTRCK47ETT7H","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":"2ea27561d3542c3060716d299e3f64f73d2d1965c4c8ae3db7f571339cbc9530","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2022-02-11T08:43:53Z","title_canon_sha256":"8c4cd546d1310ab5fcaac38cb100fffc71a566e3f838bcb11a539990ed89e8b5"},"schema_version":"1.0","source":{"id":"2202.05507","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2202.05507","created_at":"2026-07-05T08:06:10Z"},{"alias_kind":"arxiv_version","alias_value":"2202.05507v2","created_at":"2026-07-05T08:06:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2202.05507","created_at":"2026-07-05T08:06:10Z"},{"alias_kind":"pith_short_12","alias_value":"KD3CSBJKVC4T","created_at":"2026-07-05T08:06:10Z"},{"alias_kind":"pith_short_16","alias_value":"KD3CSBJKVC4TN4BT","created_at":"2026-07-05T08:06:10Z"},{"alias_kind":"pith_short_8","alias_value":"KD3CSBJK","created_at":"2026-07-05T08:06:10Z"}],"graph_snapshots":[{"event_id":"sha256:94cd3c1f9d17f2ca7b6b534fe211bdd2e167a516b100c4bbdc9eaf943038c3db","target":"graph","created_at":"2026-07-05T08:06:10Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2202.05507/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"For positive integers $s$, $t$, $m$ and $n$, the Zarankiewicz number $Z_{s,t}(m,n)$ is defined to be the maximum number of edges in a bipartite graph with parts of sizes $m$ and $n$ that has no complete biparitite subgraph containing $s$ vertices in the part of size $m$ and $t$ vertices in the part of size $n$. A simple argument shows that, for each $t \\geq 2$, $Z_{2,t}(m,n)=(t-1)\\binom{m}{2}+n$ when $n \\geq (t-1)\\binom{m}{2}$. Here, for large $m$, we determine the exact value of $Z_{2,t}(m,n)$ in almost all of the remaining cases where $n=\\Theta(tm^2)$. We establish a new family of upper boun","authors_text":"Adam Mammoliti, Daniel Horsley, Guangzhou Chen","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2022-02-11T08:43:53Z","title":"Exact values for unbalanced Zarankiewicz numbers"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2202.05507","kind":"arxiv","version":2},"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":"sha256:980417d6019a6f27e890a2a80787dcc2eb5b49555068483ceec6a043132a95ef","target":"record","created_at":"2026-07-05T08:06:10Z","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":"2ea27561d3542c3060716d299e3f64f73d2d1965c4c8ae3db7f571339cbc9530","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2022-02-11T08:43:53Z","title_canon_sha256":"8c4cd546d1310ab5fcaac38cb100fffc71a566e3f838bcb11a539990ed89e8b5"},"schema_version":"1.0","source":{"id":"2202.05507","kind":"arxiv","version":2}},"canonical_sha256":"50f629052aa8b936f0338895cf9273f9f10c7e4262f805b94b2425e9848332ac","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"50f629052aa8b936f0338895cf9273f9f10c7e4262f805b94b2425e9848332ac","first_computed_at":"2026-07-05T08:06:10.248453Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T08:06:10.248453Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Wz7UPHf0jD9xYL5oN3lQW2WyidPh+/0d2oYNBaY8o7p0H9DqbxPSsKdlRsa7hQCR74qRe7u15+Eqxxf2qsSJDw==","signature_status":"signed_v1","signed_at":"2026-07-05T08:06:10.248855Z","signed_message":"canonical_sha256_bytes"},"source_id":"2202.05507","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:980417d6019a6f27e890a2a80787dcc2eb5b49555068483ceec6a043132a95ef","sha256:94cd3c1f9d17f2ca7b6b534fe211bdd2e167a516b100c4bbdc9eaf943038c3db"],"state_sha256":"bfe7d7f770eddf7a10dd334703c4896f4e9ed30792048c44a964d55e8bab6a42"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ztPW2dZPygV6XwVpLQRpYVLpR4qZ9Yt+NJa3qx9vVMfR+ZEr4LwVlxe/98ULfjQzDn+BhFkCCpgu9sAlv0wgAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-11T18:25:23.587701Z","bundle_sha256":"f4809aea8f355a954f7479d50ef0d2a493bcdd5343b6df3d144dd8acc72aed47"}}