{"paper":{"title":"On a conjecture of Erd\\H{o}s and Szekeres","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Georgios Vlachos","submitted_at":"2015-05-28T04:59:37Z","abstract_excerpt":"Let f(n) denote the smallest positive integer such that every set of $f(n)$ points in general position in the Euclidean plane contains a convex n-gon. In a seminal paper published in 1935, Erd\\H{o}s and Szekeres proved that f(n) exists and provided an upper bound. In 1961, they also proved a lower bound, which they conjectured is optimal. Their bounds are: $2^{n-2}+1 \\leq f(n) \\leq {2n - 4 \\choose n-2}+1$. Since then, the upper bound has been improved by rougly a factor of 2, to $f(n) \\leq {2n - 5 \\choose n-2}+1$. In the current paper, we further improve the upper bound by proving that: $$ \\li"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.07549","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"}