Olsonʹs constant for the group Zp⊕Zp

Let G be a finite abelian group. By Ol(G), we mean the smallest integer t such that every subset A⊂G of cardinality t contains a non-empty subset whose sum is zero. In this article, we shall prove that for all primes p>4.67×1034, we have Ol(Zp⊕Zp)=p+Ol(Zp)−1 and hence we have Ol(Zp⊕Zp)⩽p−1+⌈2p+5 log p⌉. This, in particular, proves that a conjecture of Erdős (stated below) is true for the group Zp⊕Zp for all primes p>4.67×1034.