On zero-sum subsequences of length \(k\exp(G)\). II (Q2068607)

Let \(G\) be an additive finite abelian group of exponent \(\exp(G)\) and let \(k\) be a positive integer. A sequence over \(G\) is a finite unordered sequence with terms from \(G\) and repetition allowed. We define \(\mathsf s_{k \exp(G)}(G)\) to be the smallest integer \(t\) such that every sequence over \(G\) of length \(t\) has a zero-sum subsequence of length \(k \exp(G)\) and \(\eta_{k \exp(G)}(G)\) to be the smallest integer \(t\) such that every sequence over \(G\) of length \(t\) has a zero-sum subsequence of length between \(1\) and \(k \exp(G)\). In particular, we denote \(\mathsf s(G)=\mathsf s_{\exp(G)}(G)\) and \(\eta(G)=\eta_{\exp(G)}(G)\). It is conjectured by the first author that \(\mathsf s(G)=\eta(G)+\exp(G)-1\). Recently, this conjecture is generalized by the first author et al. to a common one \(\mathsf s_{k \exp(G)}(G) = \eta_{k \exp(G)}(G) + k \exp(G)-1\). In this paper, the authors investigated the common conjecture and obtained the direct theorems (see Theorems 1.4 and 1.5) and the associated inverse theorems (see Theorems 1.7, 1.9, and 1.10). For Part I, see [the first author et al., J. Comb. Theory, Ser. A 125, 240--253 (2014; Zbl 1302.11009)].