On four colored sets with nondecreasing diameter and the Erdos-Ginzburg-ZivTheorem

摘要

A set X, with a coloring Delta: X --> Z(m), is 7ero-sum if Sigma(xis an element ofX)Delta(x) = 0. Let f(m, r) (let f(zs)(m,2r)) 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) n7-element subsets B, and B,, not necessarily the same color, such that (a) max(B-1) - min(B-1) less than or equal to max(B-2) - min(B-2), and (b) max(B-1) < min(B-2). We show that f(zs)(m,4) = f(m,4). (C) 2002 Elsevier Science (USA). [References: 17]