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Given any integers n\\ge k\\ge 2 and q\\ge 2, what is the smallest integer N with the property that no matter how we color all k-element subsets of [N]=\\{1,2,..., N\\} with q colors, we can always find a monochromatic monotone path of length n? Denoting this minimum by N_k(q,n), it follows from the seminal 1935 paper of Erd\\H os and Szekeres that N_2(q,n)=(n-1)^q+1 and N_3(2,n) = {2n -4\\choose n-2} + 1. Determining the other v"},"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":"1105.2097","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-05-11T03:27:20Z","cross_cats_sorted":[],"title_canon_sha256":"6234c494855e1e648cb769b9b88d7031970874e11a4c528a4b0db128165361bd","abstract_canon_sha256":"64bdc5c3950a7b7f329bd8c87953ab10edb1ad4326ba58ca4b302fdac0b28491"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:58:00.943206Z","signature_b64":"AT+Deoz5E1d1a3RuETYcpBT48N3irSEW11PkdDW5xz018XeKN71STaM6x28fX9uMV1gx9rLCG4gi2r61Bbq5Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"58124d374d07d13ca097eaad8a004a0d6b876b634d41bfba8a5c91a08e8956f8","last_reissued_at":"2026-05-18T02:58:00.942565Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:58:00.942565Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Erdos-Szekeres-type theorems for monotone paths and convex bodies","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Andrew Suk, Benny Sudakov, Jacob Fox, Janos Pach","submitted_at":"2011-05-11T03:27:20Z","abstract_excerpt":"For any sequence of positive integers j_1 < j_2 < ... < j_n, the k-tuples (j_i,j_{i + 1},...,j_{i + k-1}), i=1, 2,..., n - k+1, are said to form a monotone path of length n. Given any integers n\\ge k\\ge 2 and q\\ge 2, what is the smallest integer N with the property that no matter how we color all k-element subsets of [N]=\\{1,2,..., N\\} with q colors, we can always find a monochromatic monotone path of length n? Denoting this minimum by N_k(q,n), it follows from the seminal 1935 paper of Erd\\H os and Szekeres that N_2(q,n)=(n-1)^q+1 and N_3(2,n) = {2n -4\\choose n-2} + 1. 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