Let G be an additive finite abelian group and let k (G), D (G) -1 be a positive integer. Denote by s ₊ (G) the smallest positive integer l N \+\ such that each sequence of length l over G has a non-empty zero-sum subsequence of length at most k. Let kG (G), D (G) -1 be the smallest positive integer such that s ₃ (₆) -₃ (G) D (G) +d for D (G) -d kG. We conjecture that kG=D (G) +12 for finite abelian groups G with r (G) 2 and D (G) =D^* (G). In this paper, we mainly study this conjecture for finite abelian p-groups and get some results to support this conjecture. We also prove that kG D (G) -2 for all finite abelian groups G with r (G) 2 except C₂³ and C₂⁴. In addition, we also get some lower bounds for the invariant s ₊ (G).
Kevin Zhao (Thu,) studied this question.