On Four Colored Sets with Nondecreasing Diameter and the Erdős–Ginzburg–Ziv Theorem

A set X, with a coloring Δ:X→Zm, is zero-sum if ∑x∈XΔ(x)=0. Let f(m,r) (let fzs(m,2r)) be the least N such that for every coloring of 1,…,N with r colors (with elements from r disjoint copies of Zm) there exist monochromatic (zero-sum) m-element subsets B1 and B2, not necessarily the same color, such that (a) max(B1)−min(B1)⩽max(B2)−min(B2), and (b) max(B1)<min(B2). We show that fzs(m,4)=f(m,4).