{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:6CSTHE6HFS6VYA2KNCDVTS6WPL","short_pith_number":"pith:6CSTHE6H","schema_version":"1.0","canonical_sha256":"f0a53393c72cbd5c034a688759cbd67ad3d29082779234a2d344da25d7b2dfc0","source":{"kind":"arxiv","id":"1810.04980","version":1},"attestation_state":"computed","paper":{"title":"Rainbow triangles and cliques in edge-colored graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Elena Mohr, Stefan Ehard","submitted_at":"2018-10-11T12:31:48Z","abstract_excerpt":"For an edge-colored graph, a subgraph is called rainbow if all its edges have distinct colors. We show that if $G$ is an edge-colored graph of order $n$ and size $m$ using $c$ colors on its edges, and $m+c\\geq \\binom{n+1}{2}+k-1$ for a non-negative integer $k$, then $G$ contains at least $k$ rainbow triangles. For $n\\geq 3k$, we show that this result is best possible, and we completely characterize the class of edge-colored graphs for which this result is sharp. Furthermore, we show that an edge-colored graph $G$ contains at least $k$ rainbow triangles if $\\sum\\limits_{v\\in V(G)} d^c_G(v)\\geq "},"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":"1810.04980","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-10-11T12:31:48Z","cross_cats_sorted":[],"title_canon_sha256":"12b10a264d5d9532b01e2f33bd5491f4872325f7c2b84562a3fb6c9693ac7c88","abstract_canon_sha256":"49b62ff27cd8cdfa0cbf3f325b069452a8b33f288d42517d5674ab38d8511043"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:03:35.357915Z","signature_b64":"7fTk6F8VHf5+R6y/w34wwSRSCV2Hq/6UxdTZ8r8GYv6nUNVGxm8/zpeDLxcWxkThdGdXyR+dOV+aYvR9MLA4Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f0a53393c72cbd5c034a688759cbd67ad3d29082779234a2d344da25d7b2dfc0","last_reissued_at":"2026-05-18T00:03:35.357331Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:03:35.357331Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Rainbow triangles and cliques in edge-colored graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Elena Mohr, Stefan Ehard","submitted_at":"2018-10-11T12:31:48Z","abstract_excerpt":"For an edge-colored graph, a subgraph is called rainbow if all its edges have distinct colors. We show that if $G$ is an edge-colored graph of order $n$ and size $m$ using $c$ colors on its edges, and $m+c\\geq \\binom{n+1}{2}+k-1$ for a non-negative integer $k$, then $G$ contains at least $k$ rainbow triangles. For $n\\geq 3k$, we show that this result is best possible, and we completely characterize the class of edge-colored graphs for which this result is sharp. Furthermore, we show that an edge-colored graph $G$ contains at least $k$ rainbow triangles if $\\sum\\limits_{v\\in V(G)} d^c_G(v)\\geq "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.04980","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":"1810.04980","created_at":"2026-05-18T00:03:35.357429+00:00"},{"alias_kind":"arxiv_version","alias_value":"1810.04980v1","created_at":"2026-05-18T00:03:35.357429+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1810.04980","created_at":"2026-05-18T00:03:35.357429+00:00"},{"alias_kind":"pith_short_12","alias_value":"6CSTHE6HFS6V","created_at":"2026-05-18T12:32:08.215937+00:00"},{"alias_kind":"pith_short_16","alias_value":"6CSTHE6HFS6VYA2K","created_at":"2026-05-18T12:32:08.215937+00:00"},{"alias_kind":"pith_short_8","alias_value":"6CSTHE6H","created_at":"2026-05-18T12:32:08.215937+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/6CSTHE6HFS6VYA2KNCDVTS6WPL","json":"https://pith.science/pith/6CSTHE6HFS6VYA2KNCDVTS6WPL.json","graph_json":"https://pith.science/api/pith-number/6CSTHE6HFS6VYA2KNCDVTS6WPL/graph.json","events_json":"https://pith.science/api/pith-number/6CSTHE6HFS6VYA2KNCDVTS6WPL/events.json","paper":"https://pith.science/paper/6CSTHE6H"},"agent_actions":{"view_html":"https://pith.science/pith/6CSTHE6HFS6VYA2KNCDVTS6WPL","download_json":"https://pith.science/pith/6CSTHE6HFS6VYA2KNCDVTS6WPL.json","view_paper":"https://pith.science/paper/6CSTHE6H","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1810.04980&json=true","fetch_graph":"https://pith.science/api/pith-number/6CSTHE6HFS6VYA2KNCDVTS6WPL/graph.json","fetch_events":"https://pith.science/api/pith-number/6CSTHE6HFS6VYA2KNCDVTS6WPL/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/6CSTHE6HFS6VYA2KNCDVTS6WPL/action/timestamp_anchor","attest_storage":"https://pith.science/pith/6CSTHE6HFS6VYA2KNCDVTS6WPL/action/storage_attestation","attest_author":"https://pith.science/pith/6CSTHE6HFS6VYA2KNCDVTS6WPL/action/author_attestation","sign_citation":"https://pith.science/pith/6CSTHE6HFS6VYA2KNCDVTS6WPL/action/citation_signature","submit_replication":"https://pith.science/pith/6CSTHE6HFS6VYA2KNCDVTS6WPL/action/replication_record"}},"created_at":"2026-05-18T00:03:35.357429+00:00","updated_at":"2026-05-18T00:03:35.357429+00:00"}