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We say that $A$ has a pattern ${\\cal A}$ provided that $a_{ij}=a_{i'j'}$ if and only if $(i,j),(i',j')\\in A_t$ for some $t\\in\\{1,\\ldots,r\\}$. In this note we study the following function $f$ defined on the set of all $m\\times n$ matrices $M$ with $s$ distinct entries: $f(M; {\\cal A})$ is the smallest number of positions where the entries of $M$ need to be changed such that the resulting matrix does not have any submatrix with pat"},"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":"1605.06577","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-05-21T02:50:35Z","cross_cats_sorted":[],"title_canon_sha256":"2d1bcfcc7ef403336b4db297a83121cb7140080bcfc264e51da19e36bffd95f6","abstract_canon_sha256":"c76e54f0eb79b6fa9985a4b8efcb05ef24ef07b08547054d9a19836ca2abb4f9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:13:46.744255Z","signature_b64":"k+c6RtDT/5F7ury5l8z0jsgwXAsd9bwCJcxV0U3izP+AIUxvOyRfXqgzvDNGe7pj+b9jzkZ9N2/UNZVTUa2qCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7acc21034ab984ebb23f87a2d5d20331a3f9231a882ecbef0fd40e9daae9cacd","last_reissued_at":"2026-05-18T01:13:46.743552Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:13:46.743552Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Avoiding patterns in matrices via a small number of changes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Maria Axenovich, Ryan R. 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In this note we study the following function $f$ defined on the set of all $m\\times n$ matrices $M$ with $s$ distinct entries: $f(M; {\\cal A})$ is the smallest number of positions where the entries of $M$ need to be changed such that the resulting matrix does not have any submatrix with pat"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.06577","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":""},"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":"1605.06577","created_at":"2026-05-18T01:13:46.743659+00:00"},{"alias_kind":"arxiv_version","alias_value":"1605.06577v2","created_at":"2026-05-18T01:13:46.743659+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1605.06577","created_at":"2026-05-18T01:13:46.743659+00:00"},{"alias_kind":"pith_short_12","alias_value":"PLGCCA2KXGCO","created_at":"2026-05-18T12:30:39.010887+00:00"},{"alias_kind":"pith_short_16","alias_value":"PLGCCA2KXGCOXMR7","created_at":"2026-05-18T12:30:39.010887+00:00"},{"alias_kind":"pith_short_8","alias_value":"PLGCCA2K","created_at":"2026-05-18T12:30:39.010887+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/PLGCCA2KXGCOXMR7Q6RNLUQDGG","json":"https://pith.science/pith/PLGCCA2KXGCOXMR7Q6RNLUQDGG.json","graph_json":"https://pith.science/api/pith-number/PLGCCA2KXGCOXMR7Q6RNLUQDGG/graph.json","events_json":"https://pith.science/api/pith-number/PLGCCA2KXGCOXMR7Q6RNLUQDGG/events.json","paper":"https://pith.science/paper/PLGCCA2K"},"agent_actions":{"view_html":"https://pith.science/pith/PLGCCA2KXGCOXMR7Q6RNLUQDGG","download_json":"https://pith.science/pith/PLGCCA2KXGCOXMR7Q6RNLUQDGG.json","view_paper":"https://pith.science/paper/PLGCCA2K","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1605.06577&json=true","fetch_graph":"https://pith.science/api/pith-number/PLGCCA2KXGCOXMR7Q6RNLUQDGG/graph.json","fetch_events":"https://pith.science/api/pith-number/PLGCCA2KXGCOXMR7Q6RNLUQDGG/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/PLGCCA2KXGCOXMR7Q6RNLUQDGG/action/timestamp_anchor","attest_storage":"https://pith.science/pith/PLGCCA2KXGCOXMR7Q6RNLUQDGG/action/storage_attestation","attest_author":"https://pith.science/pith/PLGCCA2KXGCOXMR7Q6RNLUQDGG/action/author_attestation","sign_citation":"https://pith.science/pith/PLGCCA2KXGCOXMR7Q6RNLUQDGG/action/citation_signature","submit_replication":"https://pith.science/pith/PLGCCA2KXGCOXMR7Q6RNLUQDGG/action/replication_record"}},"created_at":"2026-05-18T01:13:46.743659+00:00","updated_at":"2026-05-18T01:13:46.743659+00:00"}