On the Erdös-Ginzburg-Ziv Problem in large dimension

The Erdös-Ginzburg-Ziv Problem is a classical extremal problem in discrete geometry. Given positive integers m and n, the problem asks about the smallest number s such that among any s points in the integer lattice Z^n one can find m points whose centroid is again a lattice point. Despite of a lot of attention over the last 50 years, this problem is still wide open. For fixed dimension n, Alon and Dubiner proved that the answer grows linearly with m. In this talk, we discuss new bounds for the opposite case, where the number m is fixed and the dimension n is large. Joint work with Dmitrii Zakharov.