{"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. Determining the other v"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1105.2097","kind":"arxiv","version":1},"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"}