On two questions about restricted sumsets in finite abelian groups

Let G be an abelian group of finite order n, and let h be a positive integer. A subset A of G is called weakly h-incomplete if not every element of G can be written as the sum of h distinct elements of A; in particular, if A does not contain h distinct elements that add to zero, then A is called weakly h-zero-sum-free. We investigate the maximum size of weakly hincomplete and weakly h-zero-sum-free sets in G, denoted by Ch(G) and Zh(G), respectively. Among our results are the following: (i) IfG is of odd order and (n− 1)/2 ≤ h ≤ n− 2, then Ch(G) = Zh(G) = h+1, unless G is an elementary abelian 3-group and h = n−3; (ii) If G is an elementary abelian 2-group and n/2 ≤ h ≤ n − 2, then Ch(G) = Zh(G) = h + 2, unless h = n− 4.