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We have computed the new number w(2;3,19) = 349, and we provide lower bounds for 20 <= t <= 39, where for t <= 30 we conjecture these lower bounds to be exact. The lower bounds for 24 <= t <= 30 refute the conjecture that w(2;3,t) <= t^2, and we present an improved conjecture. We also investigate regularities in the good partitions (certificates) to better understand the lower bounds.\n  Motivated by such reglarities, we introduce *palindromic van der Waerden","authors_text":"Hunter Snevily, Oliver Kullmann, Tanbir Ahmed","cross_cats":["cs.DS"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-02-26T18:30:11Z","title":"On the van der Waerden numbers w(2;3,t)"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1102.5433","kind":"arxiv","version":4},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:77af85b97939dc2c4c5d27fa64c76a77db89ca24d5538b364d9b5d8493d63557","target":"record","created_at":"2026-05-18T01:41:00Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"9515dd0f4001086a77cf2748e18a5c09e1920b809732a740469025794f1763d5","cross_cats_sorted":["cs.DS"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-02-26T18:30:11Z","title_canon_sha256":"f2fc918eedb775993dac554cf755ca76fb9d1199e955b48fb85ec8ff82402677"},"schema_version":"1.0","source":{"id":"1102.5433","kind":"arxiv","version":4}},"canonical_sha256":"bc6433e0daec9d4e206978e4be3395dd093d593f9674349727e2149c1db33db1","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bc6433e0daec9d4e206978e4be3395dd093d593f9674349727e2149c1db33db1","first_computed_at":"2026-05-18T01:41:00.387284Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:41:00.387284Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"M6nFLrQlkc5o+pLO1beKHtVZY6vzntRRZlWSdcxpwgAJl9TlgpVu7eK+UAb8gt190/0Tyx6IFgmT+qAhuVsMBg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:41:00.387739Z","signed_message":"canonical_sha256_bytes"},"source_id":"1102.5433","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:77af85b97939dc2c4c5d27fa64c76a77db89ca24d5538b364d9b5d8493d63557","sha256:dca485b8de34d9838943c8b89e161f41f1c27ac15b9bfb3749fe1d403addd22c"],"state_sha256":"5ded84587d97e804ca02ad1c130a29dc739cf214845c88d31a14e3044d93ba5d"}