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For every sequence $u$ define $\\mathit{fw}(u)$, the formation width of $u$, to be the minimum $s$ for which there exists $r$ such that there is a subsequence isomorphic to $u$ in every $(r, s)$-formation. We use $\\mathit{fw}(u)$ to prove upper bounds on $\\mathit{Ex}(u, n)$ for sequences $u$ such that $u$ contains an alternati"},"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":"1308.3810","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2013-08-17T22:36:10Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"51979747c0c4ff436207ffdca85e84b88b68acf76794c95343a885cdf49526c3","abstract_canon_sha256":"62fd377d3cbf49a95db8d70c91a0a93c91b906b1ca16aeae1467c57fe27a9c5f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:37:44.792712Z","signature_b64":"bo2ndpUHjPVNUv3eFsfOWNJAVHYiHLqQS0yHMMmOa+tWu56JrVLjIihGUEZ/CeXxJZ4aP61jmomBI2eS1B4IAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"614b0ad07ba07d367d49c988575c47fa4f9e4171e15592ef4e58db2f15b03b72","last_reissued_at":"2026-05-18T02:37:44.792178Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:37:44.792178Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Bounding sequence extremal functions with formations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"cs.DM","authors_text":"Jonathan Tidor, J.T. 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