On subsets of Abelian groups with no 3-term arithmetic progression
Journal of Combinatorial Theory Series A
On subsets of finite Abelian groups with no 3-term arithmetic progressions
Journal of Combinatorial Theory Series A
Journal of Combinatorial Theory Series A
Lower Bounds For Multidimensional Zero Sums
Combinatorica
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After giving some background on sums of residue classes we explained the following problem on multidimensional zero sums which is well known in combinatorial number theory: Let f(n,d) denote the least integer such that any choice of f(n,d) elements in ℤnd contains a subset of size n whose sum is zero. Harborth [12] proved that (n–1)2d+1 ≤f(n,d) ≤(n–1)nd+1. The lower bound follows from the example in which there are n–1 copies of each of the 2d vectors with entries 0 or 1. The upper bound follows since any set of (n–1)nd+1 vectors must contain, by the pigeonhole principle, n vectors which are equivalent modulo n.