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More precisely, we show that the minimum value of $N$ such that in any $3$-edge colouring of $K_{N,N}$ there is a monochromatic copy of $H$ is at most $\\big(3/2+o(1)\\big)|V(H)|$. In particular, we determine asymptotically the $3$-colour bipartite Ramsey number for balanced grid graphs."},"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":"1804.02451","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-04-06T20:45:59Z","cross_cats_sorted":[],"title_canon_sha256":"a951afd1269e261947397ceddb0d6537f2e7c06a8f6fd223f6f77d747d3e009d","abstract_canon_sha256":"9278a753d8a59795a13a59d220879f6ed6f2aefdc4a137aa4ad12bcdac421745"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:18:59.448708Z","signature_b64":"7xb99NMhahEpiwwXi5YXw3kt+xp8z8T7xJqfZdmWmoOBJ0WszSQ3ABfm8uKyh7pIzo71TdSZBb/DaBAHLsNPBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"64b0e40f6a99fb9c2fd24efc8fb41331eef768b12354ae148f779e14242dadec","last_reissued_at":"2026-05-18T00:18:59.448323Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:18:59.448323Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Three-colour bipartite Ramsey number for graphs with small bandwidth","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Guilherme Oliveira Mota","submitted_at":"2018-04-06T20:45:59Z","abstract_excerpt":"We estimate the $3$-colour bipartite Ramsey number for balanced bipartite graphs $H$ with small bandwidth and bounded maximum degree. 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