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All of them are improvements by one over the previously best known bounds.\n  Let e(3,k,n) denote the minimum number of edges in any triangle-free graph on n vertices without independent sets of order k. The new upper bounds on R(3,k) are obtained by completing the computation of the exact values of e(3,k,n) for all n with k <= 9 and for all n <= 33 for k = 10, and by establishing new lower bounds o"},"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":"1210.5826","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-10-22T08:06:37Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"f392b3f89bed9b5773168a354b79af0e7acf294b82c9ce202412589267d76c92","abstract_canon_sha256":"41b1ec6fde1eb5a4abcb8f72626d04943caf1659fdfdf9bdbac1619031a7b9ed"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:30:25.505365Z","signature_b64":"fhQjA6mYitZeqWavlEN2aM9RNcReCDEaed9eAwidN0CdiCauHc1SHpv8qdITqni4aG1z8l6kN7o1+sNuMhjpAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0b6580d7f9f56d5f745db65822f5e994f3f65902067d3511c69bbe1053ee3117","last_reissued_at":"2026-05-18T03:30:25.504739Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:30:25.504739Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"New Computational Upper Bounds for Ramsey Numbers R(3,k)","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Jan Goedgebeur, Stanis{\\l}aw P. Radziszowski","submitted_at":"2012-10-22T08:06:37Z","abstract_excerpt":"Using computational techniques we derive six new upper bounds on the classical two-color Ramsey numbers: R(3,10) <= 42, R(3,11) <= 50, R(3,13) <= 68, R(3,14) <= 77, R(3,15) <= 87, and R(3,16) <= 98. All of them are improvements by one over the previously best known bounds.\n  Let e(3,k,n) denote the minimum number of edges in any triangle-free graph on n vertices without independent sets of order k. 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