The EGZ-constant and short zero-sum sequences over finite abelian groups

TextLet G   be an additive finite abelian group with exponent exp⁡(G)exp⁡(G). Let η(G)η(G) be the smallest integer t such that every sequence of length t   has a nonempty zero-sum subsequence of length at most exp⁡(G)exp⁡(G). Let s(G)s(G) be the EGZ-constant of G, which is defined as the smallest integer t such that every sequence of length t   has a zero-sum subsequence of length exp⁡(G)exp⁡(G). Let p   be an odd prime. We determine η(G)η(G) for some groups G   with D(G)≤2exp⁡(G)−1D(G)≤2exp⁡(G)−1, including the p-groups of rank three and the p  -groups G=Cexp⁡(G)⊕Cpmr. We also determine s(G)s(G) for the groups G   above with more larger exponent than D(G)D(G), which confirms a conjecture by Schmid and Zhuang from 2010, where D(G)D(G) denotes the Davenport constant of G.VideoFor a video summary of this paper, please visit https://youtu.be/V6yay2i75a0.