The Erd\H{o}s-Ginzburg-Ziv constant and progression-free subsets

Ellenberg and Gijswijt gave recently a new exponential upper bound for the size of three-term arithmetic progression free sets in $({\mathbb Z_p})^n$, where $p$ is a prime. Petrov summarized their method and generalized their result to linear forms. In this short note we use Petrov's result to give new exponential upper bounds for the Erd\H{o}s-Ginzburg-Ziv constant of finite Abelian groups of high rank. Our main results depend on a conjecture about Property D.