On zero-sum subsequences of length

Let G be an additive finite abelian group of exponent exp ( G ) . For every positive integer k, let s k exp ( G ) ( G ) denote the smallest integer t such that every sequence over G of length t contains a zero-sum subsequence of length k exp ( G ) . We prove that if exp ( G ) is sufficiently larger than | G | exp ( G ) then s k exp ( G ) ( G ) = k exp ( G ) + D ( G ) − 1 for all k ⩾ 2 , where D ( G ) is the Davenport constant of G.