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## Acta Arithmetica

2015 | 168 | 3 | 289-299
Tytuł artykułu

### On a conjecture of Lemke and Kleitman

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Abstrakty
EN
Let G be a finite cyclic group of order n ≥ 2. Every sequence S over G can be written in the form $S = (n_1 g)· ... · (n_lg)$ where g ∈ G and $n_1, ..., n_l ∈ [1,ord(g)]$, and the index ind(S) of S is defined as the minimum of $(n_1 + ⋯ + n_l )/ord(g)$ over all g ∈ G with ord(g) = n. In this paper it is shown that any sequence S over G of length |S| ≥ n ≥ 5, 2 ∤ n, having an element with multiplicity at least n/3 has a subsequence T with ind(T) = 1. On the other hand, if n,d ≥ 2 are positive integers with d|n and \$n > d²(d³-d²+d+1), we provide an example of a sequence S of length |S| ≥ n having an element with multiplicity l = n/d - d(d-1) - 1 such that S has no subsequence T with ind(T) = 1, giving a general counterexample to a conjecture of Lemke and Kleitman.
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Numer
Strony
289-299
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Daty
wydano
2015
Twórcy
autor
• Sino-French Institute of Nuclear Engineering and Technology, Sun Yat-Sen University, Guangzhou 510275, P.R. China
autor
• Department of Mathematics and Statistics, Brock University, St. Catharines, ON, Canada L2S 3A1
autor
• School of Mathematics, South China Normal University, Guangzhou 510631, P.R. China
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