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On a partition analog of the Cauchy-Davenport Theorem. (English) Zbl 1102.11016

Summary: Let \(G\) be a finite abelian group, and let \(n\) be a positive integer. From the Cauchy-Davenport Theorem it follows that if \(G\) is a cyclic group of prime order, then any collection of \(n\) subsets \(A_1,A_2,\ldots,A_n\) of \(G\) satisfies \(|\sum_{i=1}^n A_i| \geq \min \{|G|,\,\sum_{i=1}^n |A_i|-n+1\}\). M. Kneser [Math. Z. 64, 429–434 (1955; Zbl 0064.04305)] generalized the Cauchy-Davenport Theorem for any abelian group. In this paper, we prove a sequence-partition analog of the Cauchy-Davenport Theorem along the lines of Kneser’s Theorem. A particular case of our theorem was proved by J. E. Olson [J. Number Theory 9, 63–70 (1977; Zbl 0351.20032)] in the context of the theorem of P. Erdős, G. Ginzburg and A. Ziv [Bull. Res. Council Israel 10F, 41–43 (1961; Zbl 0063.00009)].

MSC:

11B75 Other combinatorial number theory
05D10 Ramsey theory
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