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Concentrating on ${\\hat{r}}_c(nK_2,G_2)$ where $nK_2$ is a matching, we generalise and improve two previous results as follows. Vito, Nabila, Safitri, and Silaban obtained the exact values of ${\\hat{r}}_c(nK_2,P_3)$ for $n=2,3,4$. We determine its exact values for all positive integers $n$. Rahadjeng, Baskoro, and Assiyatun proved "},"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":"2205.03965","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2022-05-08T22:48:00Z","cross_cats_sorted":[],"title_canon_sha256":"db4841d0e6c4d9d4ba21cce28e77cfa7d6254db7675b40e00a1548c5ad0cb116","abstract_canon_sha256":"8f98d3ffc4e4e0006e34de4e0ae90b23ceaeedf005299ea6eb216da904a27ece"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T04:21:19.994830Z","signature_b64":"HodyQOdv8LtFdXvCl60sYQr8webglEOyYlhC02oYHlHDrmsVIfJQbRWvfFSRhZat1TjnGVAh6EWhNEh3m1f6AQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f0ec6636a3c649b815086ebdb2fbc596e1331171095266109e3fdcb4ebf3b288","last_reissued_at":"2026-07-05T04:21:19.994388Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T04:21:19.994388Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Connected size Ramsey numbers of matchings versus a small path or cycle","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ruyu Song, Sha Wang, Yanbo Zhang, Yixin Zhang","submitted_at":"2022-05-08T22:48:00Z","abstract_excerpt":"Given two graphs $G_1, G_2$, the connected size Ramsey number ${\\hat{r}}_c(G_1,G_2)$ is defined to be the minimum number of edges of a connected graph $G$, such that for any red-blue edge colouring of $G$, there is either a red copy of $G_1$ or a blue copy of $G_2$. 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