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