{"paper":{"title":"On generalized Erd\\H{o}s-Ginzburg-Ziv constants of $C_n^r$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Dongchun Han, Hanbin Zhang","submitted_at":"2018-09-18T06:33:25Z","abstract_excerpt":"Let $G$ be an additive finite abelian group with exponent $\\exp(G)=m$. For any positive integer $k$, the $k$-th generalized Erd\\H{o}s-Ginzburg-Ziv constant $\\mathsf s_{km}(G)$ is defined as the smallest positive integer $t$ such that every sequence $S$ in $G$ of length at least $t$ has a zero-sum subsequence of length $km$. It is easy to see that $\\mathsf s_{kn}(C_n^r)\\ge(k+r)n-r$ where $n,r\\in\\mathbb N$. Kubertin conjectured that the equality holds for any $k\\ge r$. In this paper, we mainly prove the following results:\n  (1) For every positive integer $k\\ge 6$, we have $$\\mathsf s_{kn}(C_n^3)"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.06548","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"}