A variant of Kemnitz Conjecture

For any integer n⩾3, by g(Zn⊕Zn) we denote the smallest positive integer t such that every subset of cardinality t of the group Zn⊕Zn contains a subset of cardinality n whose sum is zero. Kemnitz (Extremalprobleme für Gitterpunkte, Ph.D. Thesis, Technische Universität Braunschweig, 1982) proved that g(Zp⊕Zp)=2p−1 for p=3,5,7. In this paper, as our main result, we prove that g(Zp⊕Zp)=2p−1 for all primes p⩾67.