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The extremal number of $A$, denoted by $ex(n,A)$, is the maximum number of $1$-entries in an $n\\times n$ sized matrix $M$ that does not contain $A$.\n  A matrix $A$ is column-$t$-partite (or row-$t$-partite), if it can be cut along the columns (or rows) into $t$ submatrices such that every row (or column) of these submatrices contains at most one $1$-entry. We prove that if $A$ is column-$t$-partite, then $ex(n,A)<n^{2"},"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":"1908.03189","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-08-08T17:47:58Z","cross_cats_sorted":[],"title_canon_sha256":"909564706b2085a7a82ad1f759c50429a22e188530bfa8aed38e7399cb043510","abstract_canon_sha256":"c63e2835d9fc3246e27cc3a123eaa06716dd81f16b21008ea2187b100d6f4ad7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T23:52:27.098074Z","signature_b64":"D5S3RkpAtKZMeijX6sGxkAfllyB56IGZzJm/LjLNGB56ySQz7+WJxjAl/9oq3q5ojdsqVY4utWmxvBROknKgDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f38d86246401a1ab0950d1f39df3fd125dae5208b0ca87d1e8235a5d3b7b4f6e","last_reissued_at":"2026-07-04T23:52:27.097666Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T23:52:27.097666Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Bipartite Tur\\'an problems for ordered graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Abhishek Methuku, Istv\\'an Tomon","submitted_at":"2019-08-08T17:47:58Z","abstract_excerpt":"A zero-one matrix $M$ contains a zero-one matrix $A$ if one can delete some rows and columns of $M$, and turn some 1-entries into 0-entries such that the resulting matrix is $A$. The extremal number of $A$, denoted by $ex(n,A)$, is the maximum number of $1$-entries in an $n\\times n$ sized matrix $M$ that does not contain $A$.\n  A matrix $A$ is column-$t$-partite (or row-$t$-partite), if it can be cut along the columns (or rows) into $t$ submatrices such that every row (or column) of these submatrices contains at most one $1$-entry. We prove that if $A$ is column-$t$-partite, then $ex(n,A)<n^{2"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.03189","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1908.03189/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1908.03189","created_at":"2026-07-04T23:52:27.097733+00:00"},{"alias_kind":"arxiv_version","alias_value":"1908.03189v1","created_at":"2026-07-04T23:52:27.097733+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.03189","created_at":"2026-07-04T23:52:27.097733+00:00"},{"alias_kind":"pith_short_12","alias_value":"6OGYMJDEAGQ2","created_at":"2026-07-04T23:52:27.097733+00:00"},{"alias_kind":"pith_short_16","alias_value":"6OGYMJDEAGQ2WCKQ","created_at":"2026-07-04T23:52:27.097733+00:00"},{"alias_kind":"pith_short_8","alias_value":"6OGYMJDE","created_at":"2026-07-04T23:52:27.097733+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/6OGYMJDEAGQ2WCKQ2HZZ3475CJ","json":"https://pith.science/pith/6OGYMJDEAGQ2WCKQ2HZZ3475CJ.json","graph_json":"https://pith.science/api/pith-number/6OGYMJDEAGQ2WCKQ2HZZ3475CJ/graph.json","events_json":"https://pith.science/api/pith-number/6OGYMJDEAGQ2WCKQ2HZZ3475CJ/events.json","paper":"https://pith.science/paper/6OGYMJDE"},"agent_actions":{"view_html":"https://pith.science/pith/6OGYMJDEAGQ2WCKQ2HZZ3475CJ","download_json":"https://pith.science/pith/6OGYMJDEAGQ2WCKQ2HZZ3475CJ.json","view_paper":"https://pith.science/paper/6OGYMJDE","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1908.03189&json=true","fetch_graph":"https://pith.science/api/pith-number/6OGYMJDEAGQ2WCKQ2HZZ3475CJ/graph.json","fetch_events":"https://pith.science/api/pith-number/6OGYMJDEAGQ2WCKQ2HZZ3475CJ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/6OGYMJDEAGQ2WCKQ2HZZ3475CJ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/6OGYMJDEAGQ2WCKQ2HZZ3475CJ/action/storage_attestation","attest_author":"https://pith.science/pith/6OGYMJDEAGQ2WCKQ2HZZ3475CJ/action/author_attestation","sign_citation":"https://pith.science/pith/6OGYMJDEAGQ2WCKQ2HZZ3475CJ/action/citation_signature","submit_replication":"https://pith.science/pith/6OGYMJDEAGQ2WCKQ2HZZ3475CJ/action/replication_record"}},"created_at":"2026-07-04T23:52:27.097733+00:00","updated_at":"2026-07-04T23:52:27.097733+00:00"}