{"paper":{"title":"Erd\\H{o}s-Szekeres without induction","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Sergey Norin, Yelena Yuditsky","submitted_at":"2015-09-10T20:49:54Z","abstract_excerpt":"Let $ES(n)$ be the minimal integer such that any set of $ES(n)$ points in the plane in general position contains $n$ points in convex position. The problem of estimating $ES(n)$ was first formulated by Erd\\H{o}s and Szekeres, who proved that $ES(n) \\leq \\binom{2n-4}{n-2}+1$. The current best upper bound, $\\lim\\sup_{n \\to \\infty} \\frac{ES(n)}{\\binom{2n-5}{n-2}}\\le \\frac{29}{32}$, is due to Vlachos. We improve this to $$\\lim\\sup_{n \\to \\infty} \\frac{ES(n)}{\\binom{2n-5}{n-2}}\\le \\frac{7}{8}.$$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.03332","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"}