{"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"}