{"paper":{"title":"On sequences covering all rainbow $k$-progressions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Leonardo Alese, Paul Tabatabai, Stefan Lendl","submitted_at":"2018-02-09T14:48:12Z","abstract_excerpt":"Let $\\text{ac}(n,k)$ denote the smallest positive integer with the property that there exists an $n$-colouring $f$ of $\\{1,\\dots,\\text{ac}(n,k)\\}$ such that for every $k$-subset $R \\subseteq \\{1, \\dots, n\\}$ there exists an (arithmetic) $k$-progression $A$ in $\\{1,\\dots,\\text{ac}(n,k)\\}$ with $\\{f(a) : a \\in A\\} = R$. Determining the behaviour of the function $\\text{ac}(n,k)$ is a previously unstudied problem. We use the first moment method to give an asymptotic upper bound for $\\text{ac}(n,k)$ for the case $k = o(n^{1/{5}})$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1802.03285","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"}