In 1961, Erdos, Ginzburg and Ziv proved a classical result that any sequence of 2n-1 integers contains a subsequence of length n whose sum is divisible by n. Further studies of this problem have been an important topic in additive combinatorics.
Let G be a finite abelian group with exponent exp(G). The Erdos-Ginzburg-Ziv constant s(G) is the smallest positive integer t such that any sequence S over G contains a zero-sum subsequence T of length exp(G). The EGZ constant s(G) (and its generalized version s_k(G)) is closely related to the study of the Davenport constant, the cap set problem, and coding theory. In this talk, we introduce some recent progress on these problems. This talk is based on a series joint works with Dongchun Han, and with Shiwen Zhang.