{"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"}