{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:X4ESDPJXWLY65HUXFJQA55BQ5O","short_pith_number":"pith:X4ESDPJX","schema_version":"1.0","canonical_sha256":"bf0921bd37b2f1ee9e972a600ef430eb8370981881099840852cd0626d113d79","source":{"kind":"arxiv","id":"1509.05632","version":1},"attestation_state":"computed","paper":{"title":"Periods in missing lengths of rainbow cycles","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Petr Vojt\\v{e}chovsk\\'y","submitted_at":"2015-09-18T13:47:25Z","abstract_excerpt":"A cycle in an edge-colored graph is said to be rainbow if no two of its edges have the same color. For a complete, infinite, edge-colored graph $G$, define $\\mathfrak{S}(G)=\\{n\\ge 2\\;|\\;\\text{no $n$-cycle of $G$ is rainbow}\\}$. Then $\\mathfrak{S}(G)$ is a monoid with respect to the operation $n\\circ m = n+m-2$, and thus there is a least positive integer $\\pi(G)$, the period of $\\mathfrak{S}(G)$, such that $\\mathfrak{S}(G)$ contains the arithmetic progression $\\{N+k\\pi(G)\\;|\\;k\\ge 0\\}$ for some sufficiently large $N$.\n  Given that $n\\in\\mathfrak{S}(G)$, what can be said about $\\pi(G)$? Alexeev "},"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":"1509.05632","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-09-18T13:47:25Z","cross_cats_sorted":[],"title_canon_sha256":"88e8777356c1ea07576f46fb703d519fd9df59f6a4aa94df9123e26d3004e976","abstract_canon_sha256":"f9ebbbacbec91a609771a0ec8b0359d789b15e10366d4162580a5106d0ccbc3b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:32:42.733800Z","signature_b64":"e7mB3Ry5jFi4ZSKJkwI1SwiNfdD5lzNoM8rZVapY3KXaS1DKyVMcWBbRMus/m9L0zsAPmLpH94shNvWj4x9OCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bf0921bd37b2f1ee9e972a600ef430eb8370981881099840852cd0626d113d79","last_reissued_at":"2026-05-18T01:32:42.733259Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:32:42.733259Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Periods in missing lengths of rainbow cycles","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Petr Vojt\\v{e}chovsk\\'y","submitted_at":"2015-09-18T13:47:25Z","abstract_excerpt":"A cycle in an edge-colored graph is said to be rainbow if no two of its edges have the same color. For a complete, infinite, edge-colored graph $G$, define $\\mathfrak{S}(G)=\\{n\\ge 2\\;|\\;\\text{no $n$-cycle of $G$ is rainbow}\\}$. Then $\\mathfrak{S}(G)$ is a monoid with respect to the operation $n\\circ m = n+m-2$, and thus there is a least positive integer $\\pi(G)$, the period of $\\mathfrak{S}(G)$, such that $\\mathfrak{S}(G)$ contains the arithmetic progression $\\{N+k\\pi(G)\\;|\\;k\\ge 0\\}$ for some sufficiently large $N$.\n  Given that $n\\in\\mathfrak{S}(G)$, what can be said about $\\pi(G)$? Alexeev "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.05632","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":"1509.05632","created_at":"2026-05-18T01:32:42.733328+00:00"},{"alias_kind":"arxiv_version","alias_value":"1509.05632v1","created_at":"2026-05-18T01:32:42.733328+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.05632","created_at":"2026-05-18T01:32:42.733328+00:00"},{"alias_kind":"pith_short_12","alias_value":"X4ESDPJXWLY6","created_at":"2026-05-18T12:29:47.479230+00:00"},{"alias_kind":"pith_short_16","alias_value":"X4ESDPJXWLY65HUX","created_at":"2026-05-18T12:29:47.479230+00:00"},{"alias_kind":"pith_short_8","alias_value":"X4ESDPJX","created_at":"2026-05-18T12:29:47.479230+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/X4ESDPJXWLY65HUXFJQA55BQ5O","json":"https://pith.science/pith/X4ESDPJXWLY65HUXFJQA55BQ5O.json","graph_json":"https://pith.science/api/pith-number/X4ESDPJXWLY65HUXFJQA55BQ5O/graph.json","events_json":"https://pith.science/api/pith-number/X4ESDPJXWLY65HUXFJQA55BQ5O/events.json","paper":"https://pith.science/paper/X4ESDPJX"},"agent_actions":{"view_html":"https://pith.science/pith/X4ESDPJXWLY65HUXFJQA55BQ5O","download_json":"https://pith.science/pith/X4ESDPJXWLY65HUXFJQA55BQ5O.json","view_paper":"https://pith.science/paper/X4ESDPJX","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1509.05632&json=true","fetch_graph":"https://pith.science/api/pith-number/X4ESDPJXWLY65HUXFJQA55BQ5O/graph.json","fetch_events":"https://pith.science/api/pith-number/X4ESDPJXWLY65HUXFJQA55BQ5O/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/X4ESDPJXWLY65HUXFJQA55BQ5O/action/timestamp_anchor","attest_storage":"https://pith.science/pith/X4ESDPJXWLY65HUXFJQA55BQ5O/action/storage_attestation","attest_author":"https://pith.science/pith/X4ESDPJXWLY65HUXFJQA55BQ5O/action/author_attestation","sign_citation":"https://pith.science/pith/X4ESDPJXWLY65HUXFJQA55BQ5O/action/citation_signature","submit_replication":"https://pith.science/pith/X4ESDPJXWLY65HUXFJQA55BQ5O/action/replication_record"}},"created_at":"2026-05-18T01:32:42.733328+00:00","updated_at":"2026-05-18T01:32:42.733328+00:00"}