On a partition analog of the Cauchy-Davenport theorem

Summary

Let <InlineEquation><EquationSource=”tex”>G</EquationSource></InlineEquation> be a finite abelian group, and let <InlineEquation><EquationSource=”tex”>n</EquationSource></InlineEquation> be a positive integer. From the Cauchy-Davenport Theorem it follows that if <InlineEquation><EquationSource=”tex”>G</EquationSource></InlineEquation> is a cyclic group of prime order, then any collection of <InlineEquation><EquationSource=”tex”>n</EquationSource></InlineEquation> subsets <InlineEquation><EquationSource=”tex”>A_1,A_2,\ldots,A_n</EquationSource></InlineEquation> of <InlineEquation><EquationSource=”tex”>G</EquationSource></InlineEquation> satisfies <InlineEquation><EquationSource=”tex”>\bigg|\sum_{i=1}^n A_i\bigg| \ge \min \bigg\{|G|,\,\sum_{i=1}^n |A_i|-n+1\bigg\}.</EquationSource></InlineEquation> M.~Kneser 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&apos;s Theorem. A particular case of our theorem was proved by J.~E. Olson in the context of the Erd&odblac;s--Ginzburg--Ziv Theorem.