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In this paper, we determine the exact value of $r(k,l,m)$ completely. Applying a technique originated by {\\L}uczak that applies Szemer\\'edi's Regularity Lemma to reduce the problem of showing the existence of a monochromatic cycle to show the existence of a monochromatic matching in a component, we obta"},"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":"1809.05413","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-09-13T12:01:38Z","cross_cats_sorted":[],"title_canon_sha256":"c279d22b9047503e6b8ed22c3601bdaa3652a1b59e1057901a6b3ff1053a93d6","abstract_canon_sha256":"d532315325dc98916c099e44e61b67df58fc1cbe96028b6e62bd891b323ca823"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:05:27.497252Z","signature_b64":"i5G1BcEKNv9M1XmpVTPzFOf1LFddu+rIH4OAzWoEWt8N3ynIWBavLepyett+Pnq5nEweUmuYI3qDBJVtYIz8DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fcb025da2463e639864f1b96267a11d1e7cc38de00d725b203d8079754fba8e6","last_reissued_at":"2026-05-18T00:05:27.496615Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:05:27.496615Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"3-colored asymmetric bipartite Ramsey number of connected matchings and cycles","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Yuejian Peng, Zhidan Luo","submitted_at":"2018-09-13T12:01:38Z","abstract_excerpt":"Let $k,l,m$ be integers and $r(k,l,m)$ be the minimum integer $N$ such that for any red-blue-green coloring of $K_{N,N}$, there is a red matching of size at least $k$ in a component, or a blue matching of at least size $l$ in a component, or a green matching of size at least $m$ in a component. 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