{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:WTUJYHBU522MJDX765GVZQE47T","short_pith_number":"pith:WTUJYHBU","schema_version":"1.0","canonical_sha256":"b4e89c1c34eeb4c48efff74d5cc09cfcc5d49d14581255c2136ec1826c5c60ba","source":{"kind":"arxiv","id":"1605.05074","version":2},"attestation_state":"computed","paper":{"title":"The Erd\\H{o}s-Rothschild problem on edge-colourings with forbidden monochromatic cliques","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Katherine Staden, Oleg Pikhurko, Zelealem B. Yilma","submitted_at":"2016-05-17T09:40:00Z","abstract_excerpt":"Let $\\mathbf{k} := (k_1,\\dots,k_s)$ be a sequence of natural numbers. For a graph $G$, let $F(G;\\mathbf{k})$ denote the number of colourings of the edges of $G$ with colours $1,\\dots,s$ such that, for every $c \\in \\{1,\\dots,s\\}$, the edges of colour $c$ contain no clique of order $k_c$. Write $F(n;\\mathbf{k})$ to denote the maximum of $F(G;\\mathbf{k})$ over all graphs $G$ on $n$ vertices. This problem was first considered by Erd\\H{o}s and Rothschild in 1974, but it has been solved only for a very small number of non-trivial cases.\n  We prove that, for every $\\mathbf{k}$ and $n$, there is a com"},"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":"1605.05074","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-05-17T09:40:00Z","cross_cats_sorted":[],"title_canon_sha256":"78518252d5b6905453729e196b21fa8a80d8ecd8e1dd62306049f9deabad46d5","abstract_canon_sha256":"f6b4efd358501cb01347091c4aeafff254c7d954497eb5f5933eae903704a217"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:33:23.696071Z","signature_b64":"itgMeujkXhON9NehAJeAjdzwnRdZVLQCwoI8lMkQAmzMNXYXG7vEJnyhWbL8VwGr7IuxxZ+oSXzX/XFxc3W/BQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b4e89c1c34eeb4c48efff74d5cc09cfcc5d49d14581255c2136ec1826c5c60ba","last_reissued_at":"2026-05-18T00:33:23.695447Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:33:23.695447Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Erd\\H{o}s-Rothschild problem on edge-colourings with forbidden monochromatic cliques","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Katherine Staden, Oleg Pikhurko, Zelealem B. Yilma","submitted_at":"2016-05-17T09:40:00Z","abstract_excerpt":"Let $\\mathbf{k} := (k_1,\\dots,k_s)$ be a sequence of natural numbers. For a graph $G$, let $F(G;\\mathbf{k})$ denote the number of colourings of the edges of $G$ with colours $1,\\dots,s$ such that, for every $c \\in \\{1,\\dots,s\\}$, the edges of colour $c$ contain no clique of order $k_c$. Write $F(n;\\mathbf{k})$ to denote the maximum of $F(G;\\mathbf{k})$ over all graphs $G$ on $n$ vertices. This problem was first considered by Erd\\H{o}s and Rothschild in 1974, but it has been solved only for a very small number of non-trivial cases.\n  We prove that, for every $\\mathbf{k}$ and $n$, there is a com"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.05074","kind":"arxiv","version":2},"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":"1605.05074","created_at":"2026-05-18T00:33:23.695565+00:00"},{"alias_kind":"arxiv_version","alias_value":"1605.05074v2","created_at":"2026-05-18T00:33:23.695565+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1605.05074","created_at":"2026-05-18T00:33:23.695565+00:00"},{"alias_kind":"pith_short_12","alias_value":"WTUJYHBU522M","created_at":"2026-05-18T12:30:51.357362+00:00"},{"alias_kind":"pith_short_16","alias_value":"WTUJYHBU522MJDX7","created_at":"2026-05-18T12:30:51.357362+00:00"},{"alias_kind":"pith_short_8","alias_value":"WTUJYHBU","created_at":"2026-05-18T12:30:51.357362+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/WTUJYHBU522MJDX765GVZQE47T","json":"https://pith.science/pith/WTUJYHBU522MJDX765GVZQE47T.json","graph_json":"https://pith.science/api/pith-number/WTUJYHBU522MJDX765GVZQE47T/graph.json","events_json":"https://pith.science/api/pith-number/WTUJYHBU522MJDX765GVZQE47T/events.json","paper":"https://pith.science/paper/WTUJYHBU"},"agent_actions":{"view_html":"https://pith.science/pith/WTUJYHBU522MJDX765GVZQE47T","download_json":"https://pith.science/pith/WTUJYHBU522MJDX765GVZQE47T.json","view_paper":"https://pith.science/paper/WTUJYHBU","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1605.05074&json=true","fetch_graph":"https://pith.science/api/pith-number/WTUJYHBU522MJDX765GVZQE47T/graph.json","fetch_events":"https://pith.science/api/pith-number/WTUJYHBU522MJDX765GVZQE47T/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/WTUJYHBU522MJDX765GVZQE47T/action/timestamp_anchor","attest_storage":"https://pith.science/pith/WTUJYHBU522MJDX765GVZQE47T/action/storage_attestation","attest_author":"https://pith.science/pith/WTUJYHBU522MJDX765GVZQE47T/action/author_attestation","sign_citation":"https://pith.science/pith/WTUJYHBU522MJDX765GVZQE47T/action/citation_signature","submit_replication":"https://pith.science/pith/WTUJYHBU522MJDX765GVZQE47T/action/replication_record"}},"created_at":"2026-05-18T00:33:23.695565+00:00","updated_at":"2026-05-18T00:33:23.695565+00:00"}