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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"},"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":"1505.07549","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-05-28T04:59:37Z","cross_cats_sorted":[],"title_canon_sha256":"cf8d6ac9b0676602ac8367fcf6c3ce573cb704ac14036aeccea2df14327e900e","abstract_canon_sha256":"6f3bcb26652dff5027e225e733da9093eb5ed9501ba5d0950746f330b9ad5297"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:00:58.065757Z","signature_b64":"4nxHZfWZA4WmJL5yQRMrkmu6AgFQkiNJkatMEE+TX5iNSVHv/qFCaM8TGXXAGZhiaclTAKaG9eVr5s0crDppDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"85692b8b1c82724bbafe6903d551b981f9b8036c4308aab9004780e128d9e9ee","last_reissued_at":"2026-05-18T02:00:58.064969Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:00:58.064969Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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