{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2025:ACG7FNTJH63SUBMYPQZYKCR6K4","short_pith_number":"pith:ACG7FNTJ","schema_version":"1.0","canonical_sha256":"008df2b6693fb72a05987c33850a3e572080bc6df2eeebafd4e5364bd836f269","source":{"kind":"arxiv","id":"2510.23573","version":4},"attestation_state":"computed","paper":{"title":"An Erd\\H{o}s--Szekeres type result for words with repeats","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Every word with kn^6+1 repeats must contain one of seven specific patterns.","cross_cats":[],"primary_cat":"math.CO","authors_text":"Abigail Ollson, Jun Yan, Kyle Celano, Niraj Velankar","submitted_at":"2025-10-27T17:42:16Z","abstract_excerpt":"We prove an Erd\\H{o}s--Szekeres type result for finite words over $\\mathbb{N}$ with repeated values. Specifically, we define a \\emph{repeat} in a word to be an occurrence of a value which is not its first occurrence. We define an occurrence of a \\emph{pattern} $\\pi$ in a word $w$ to be a (not necessarily consecutive) subword of $w$ that is order isomorphic to $\\pi$. In this note, we show that every word with $kn^6+1$ repeats contains one of the following patterns: $0^{k+2}$, $0011\\cdots nn$, $nn\\cdots1100$, $012 \\cdots n012 \\cdots n$, $012 \\cdots nn\\cdots 210$, $n\\cdots 210012\\cdots n$, $n\\cdo"},"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":true},"canonical_record":{"source":{"id":"2510.23573","kind":"arxiv","version":4},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2025-10-27T17:42:16Z","cross_cats_sorted":[],"title_canon_sha256":"3bda29e12e3072a3a47d471410bb5d78e680b8d8cdbcc19cc18f4e38d9724b52","abstract_canon_sha256":"c9b591c8cda0dc28b2a84936a36a30821fd296c8adc04e928ebb60340e0664b7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-20T02:05:37.512376Z","signature_b64":"gTn5+EwlARu0xtv8RzAyYnAQIj8z7AXBaAxhy+FHANlsael4V/u4EpvHc91JzMwwqAMajYHwvMonbQqRlzHTDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"008df2b6693fb72a05987c33850a3e572080bc6df2eeebafd4e5364bd836f269","last_reissued_at":"2026-05-20T02:05:37.511543Z","signature_status":"signed_v1","first_computed_at":"2026-05-20T02:05:37.511543Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"An Erd\\H{o}s--Szekeres type result for words with repeats","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Every word with kn^6+1 repeats must contain one of seven specific patterns.","cross_cats":[],"primary_cat":"math.CO","authors_text":"Abigail Ollson, Jun Yan, Kyle Celano, Niraj Velankar","submitted_at":"2025-10-27T17:42:16Z","abstract_excerpt":"We prove an Erd\\H{o}s--Szekeres type result for finite words over $\\mathbb{N}$ with repeated values. Specifically, we define a \\emph{repeat} in a word to be an occurrence of a value which is not its first occurrence. We define an occurrence of a \\emph{pattern} $\\pi$ in a word $w$ to be a (not necessarily consecutive) subword of $w$ that is order isomorphic to $\\pi$. In this note, we show that every word with $kn^6+1$ repeats contains one of the following patterns: $0^{k+2}$, $0011\\cdots nn$, $nn\\cdots1100$, $012 \\cdots n012 \\cdots n$, $012 \\cdots nn\\cdots 210$, $n\\cdots 210012\\cdots n$, $n\\cdo"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We show that every word with kn^6+1 repeats contains one of the following patterns: 0^{k+2}, 0011⋯nn, nn⋯1100, 012⋯n012⋯n, 012⋯nn⋯210, n⋯210012⋯n, n⋯210n⋯210. Moreover, when k=1, this is best possible by constructing a word with n^6 repeats that does not contain any of these patterns.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The central claim rests on the specific choice of the seven target patterns as the complete set of unavoidable configurations once the repeat count exceeds the stated threshold; if a different or larger set of patterns were required to be avoided, the quantitative bound would not necessarily hold.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Every word with kn^6+1 repeats contains one of the patterns 0^{k+2}, 0011⋯nn, nn⋯1100, 012⋯n012⋯n, 012⋯nn⋯210, n⋯210012⋯n, or n⋯210n⋯210, with the bound tight for k=1 via an explicit construction.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Every word with kn^6+1 repeats must contain one of seven specific patterns.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"d0215262a008421df88b1f3bbda65d80f5f2af99bb438d21778d6075489bb8ba"},"source":{"id":"2510.23573","kind":"arxiv","version":4},"verdict":{"id":"5aa0ebc8-332a-420f-b240-3dd87677e71d","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-18T03:19:32.078774Z","strongest_claim":"We show that every word with kn^6+1 repeats contains one of the following patterns: 0^{k+2}, 0011⋯nn, nn⋯1100, 012⋯n012⋯n, 012⋯nn⋯210, n⋯210012⋯n, n⋯210n⋯210. Moreover, when k=1, this is best possible by constructing a word with n^6 repeats that does not contain any of these patterns.","one_line_summary":"Every word with kn^6+1 repeats contains one of the patterns 0^{k+2}, 0011⋯nn, nn⋯1100, 012⋯n012⋯n, 012⋯nn⋯210, n⋯210012⋯n, or n⋯210n⋯210, with the bound tight for k=1 via an explicit construction.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The central claim rests on the specific choice of the seven target patterns as the complete set of unavoidable configurations once the repeat count exceeds the stated threshold; if a different or larger set of patterns were required to be avoided, the quantitative bound would not necessarily hold.","pith_extraction_headline":"Every word with kn^6+1 repeats must contain one of seven specific patterns."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2510.23573/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":1,"snapshot_sha256":"98b49e42be43bbb2bf652fe7219b1367ea47be13967f957aecb0279d13d53854"},"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":"2510.23573","created_at":"2026-05-20T02:05:37.511650+00:00"},{"alias_kind":"arxiv_version","alias_value":"2510.23573v4","created_at":"2026-05-20T02:05:37.511650+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2510.23573","created_at":"2026-05-20T02:05:37.511650+00:00"},{"alias_kind":"pith_short_12","alias_value":"ACG7FNTJH63S","created_at":"2026-05-20T02:05:37.511650+00:00"},{"alias_kind":"pith_short_16","alias_value":"ACG7FNTJH63SUBMY","created_at":"2026-05-20T02:05:37.511650+00:00"},{"alias_kind":"pith_short_8","alias_value":"ACG7FNTJ","created_at":"2026-05-20T02:05:37.511650+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":1,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/ACG7FNTJH63SUBMYPQZYKCR6K4","json":"https://pith.science/pith/ACG7FNTJH63SUBMYPQZYKCR6K4.json","graph_json":"https://pith.science/api/pith-number/ACG7FNTJH63SUBMYPQZYKCR6K4/graph.json","events_json":"https://pith.science/api/pith-number/ACG7FNTJH63SUBMYPQZYKCR6K4/events.json","paper":"https://pith.science/paper/ACG7FNTJ"},"agent_actions":{"view_html":"https://pith.science/pith/ACG7FNTJH63SUBMYPQZYKCR6K4","download_json":"https://pith.science/pith/ACG7FNTJH63SUBMYPQZYKCR6K4.json","view_paper":"https://pith.science/paper/ACG7FNTJ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2510.23573&json=true","fetch_graph":"https://pith.science/api/pith-number/ACG7FNTJH63SUBMYPQZYKCR6K4/graph.json","fetch_events":"https://pith.science/api/pith-number/ACG7FNTJH63SUBMYPQZYKCR6K4/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ACG7FNTJH63SUBMYPQZYKCR6K4/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ACG7FNTJH63SUBMYPQZYKCR6K4/action/storage_attestation","attest_author":"https://pith.science/pith/ACG7FNTJH63SUBMYPQZYKCR6K4/action/author_attestation","sign_citation":"https://pith.science/pith/ACG7FNTJH63SUBMYPQZYKCR6K4/action/citation_signature","submit_replication":"https://pith.science/pith/ACG7FNTJH63SUBMYPQZYKCR6K4/action/replication_record"}},"created_at":"2026-05-20T02:05:37.511650+00:00","updated_at":"2026-05-20T02:05:37.511650+00:00"}