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The $s$-colour size-Ramsey number ${\\hat{r}}_s(H)$ of a graph $H$ is defined to be ${\\hat{r}}_s(H)=\\min\\{|E(G)|\\colon G\\rightarrow (H)_s\\}$. We prove that, for all positive integers $k$ and $s$, we have ${\\hat{r}}_s(P_n^k)=O(n)$, where $P_n^k$ is the $k$th power of the $n$-vertex path $P_n$."},"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":"1811.00844","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-11-02T13:13:07Z","cross_cats_sorted":[],"title_canon_sha256":"17328ca6549b98bf979f18c8151f48d589b4cf5d23c7244127836c40ddc96d89","abstract_canon_sha256":"74c3ecc71aa6b9dabc99970572fd6f5206e6603e840513d8227c2586ed5b74d9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:01:41.996166Z","signature_b64":"D/wSAsfbF2fefebb3cA4sLHR83gXpD8iiF6hkiqHGG7eRFse9DOhEsAnAx+6Z4tA5G2pDSETcbprhbA93LA8AA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"dfa9c7ef0b73a38c78420bb2fb7796863ac8bd2cb4a1d3b3416e1c9dbebcd5e2","last_reissued_at":"2026-05-18T00:01:41.995682Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:01:41.995682Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The multicolour size-Ramsey number of powers of paths","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Barnaby Roberts, Guilherme Oliveira Mota, Jie Han, Matthew Jenssen, Yoshiharu Kohayakawa","submitted_at":"2018-11-02T13:13:07Z","abstract_excerpt":"Given a positive integer $s$, a graph $G$ is $s$-Ramsey for a graph $H$, denoted $G\\rightarrow (H)_s$, if every $s$-colouring of the edges of $G$ contains a monochromatic copy of $H$. The $s$-colour size-Ramsey number ${\\hat{r}}_s(H)$ of a graph $H$ is defined to be ${\\hat{r}}_s(H)=\\min\\{|E(G)|\\colon G\\rightarrow (H)_s\\}$. We prove that, for all positive integers $k$ and $s$, we have ${\\hat{r}}_s(P_n^k)=O(n)$, where $P_n^k$ is the $k$th power of the $n$-vertex path $P_n$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.00844","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1811.00844","created_at":"2026-05-18T00:01:41.995758+00:00"},{"alias_kind":"arxiv_version","alias_value":"1811.00844v1","created_at":"2026-05-18T00:01:41.995758+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1811.00844","created_at":"2026-05-18T00:01:41.995758+00:00"},{"alias_kind":"pith_short_12","alias_value":"36U4P3YLOORY","created_at":"2026-05-18T12:32:02.567920+00:00"},{"alias_kind":"pith_short_16","alias_value":"36U4P3YLOORYY6CC","created_at":"2026-05-18T12:32:02.567920+00:00"},{"alias_kind":"pith_short_8","alias_value":"36U4P3YL","created_at":"2026-05-18T12:32:02.567920+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"1906.09185","citing_title":"The size Ramsey number of graphs with bounded treewidth","ref_index":25,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/36U4P3YLOORYY6CCBOZPW54WQY","json":"https://pith.science/pith/36U4P3YLOORYY6CCBOZPW54WQY.json","graph_json":"https://pith.science/api/pith-number/36U4P3YLOORYY6CCBOZPW54WQY/graph.json","events_json":"https://pith.science/api/pith-number/36U4P3YLOORYY6CCBOZPW54WQY/events.json","paper":"https://pith.science/paper/36U4P3YL"},"agent_actions":{"view_html":"https://pith.science/pith/36U4P3YLOORYY6CCBOZPW54WQY","download_json":"https://pith.science/pith/36U4P3YLOORYY6CCBOZPW54WQY.json","view_paper":"https://pith.science/paper/36U4P3YL","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1811.00844&json=true","fetch_graph":"https://pith.science/api/pith-number/36U4P3YLOORYY6CCBOZPW54WQY/graph.json","fetch_events":"https://pith.science/api/pith-number/36U4P3YLOORYY6CCBOZPW54WQY/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/36U4P3YLOORYY6CCBOZPW54WQY/action/timestamp_anchor","attest_storage":"https://pith.science/pith/36U4P3YLOORYY6CCBOZPW54WQY/action/storage_attestation","attest_author":"https://pith.science/pith/36U4P3YLOORYY6CCBOZPW54WQY/action/author_attestation","sign_citation":"https://pith.science/pith/36U4P3YLOORYY6CCBOZPW54WQY/action/citation_signature","submit_replication":"https://pith.science/pith/36U4P3YLOORYY6CCBOZPW54WQY/action/replication_record"}},"created_at":"2026-05-18T00:01:41.995758+00:00","updated_at":"2026-05-18T00:01:41.995758+00:00"}