{"paper":{"title":"An improved upper bound for the Erd\\H{o}s-Szekeres conjecture","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Georgios Vlachos, Hossein Nassajian Mojarrad","submitted_at":"2015-10-21T13:59:45Z","abstract_excerpt":"Let $ES(n)$ denote the minimum natural number such that every set of $ES(n)$ points in general position in the plane contains $n$ points in convex position. In 1935, Erd\\H{o}s and Szekeres proved that $ES(n) \\le {2n-4 \\choose n-2}+1$. In 1961, they obtained the lower bound $2^{n-2}+1 \\le ES(n)$, which they conjectured to be optimal. In this paper, we prove that $$ES(n) \\le {2n-5 \\choose n-2}-{2n-8 \\choose n-3}+2 \\approx \\frac{7}{16} {2n-4 \\choose n-2}.$$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.06255","kind":"arxiv","version":2},"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"}