 
Summary: Zerosum sets of prescribed size
Noga Alon and Moshe Dubiner
Department of Mathematics
Raymond and Beverly Sackler Faculty of Exact Sciences
Tel Aviv University, Tel Aviv, Israel
Abstract
Erdos, Ginzburg and Ziv proved that any sequence of 2n1 integers contains a subsequence of
cardinality n the sum of whose elements is divisible by n. We present several proofs of this result,
illustrating various combinatorial and algebraic tools that have numerous other applications in
Combinatorial Number Theory. Our main new results deal with an analogous multi dimensional
question. We show that any sequence of 6n  5 elements of Zn Zn contains an nsubset the
sum of whose elements is the zero vector and consider briefly the higher dimensional case as
well.
1 Introduction
The following theorem was proved in 1961 by Erdos, Ginzburg and Ziv.
Theorem 1.1 ([18]) For any sequence a1, a2, . . . , a2n1 of (not necessarily distinct) members of
the cyclic group Zn there is a set I {1, 2, . . . , 2n  1} of cardinality I = n so that iI ai = 0
(in Zn).
This theorem has motivated the recent study of certain Ramsey type problems for graphs
initiated by Bialostocki and Dierker in [8], [9] and studied by various researchers; see, e.g., [21],
