Monochromatic and Zero-Sum Sets of Nondecreasing Diameter

Abstract

Let k, r, s in the natural numbers where r \geq s \geq 2. Define f(s,r,k) to be the smallest positive integer n such that for every coloring of the integers in [1,n] there exist subsets S_1 and S_2 such that: (a) S_1 and S_2 are monochromatic (but not necessarily of the same color), (b) |S_1| = s, |S_2| = r, (c)max(S_1) < min(S_2), and (d) diam(S_1) \leq diam(S_2). We prove that the theorems defining f(s,r,2) and f(s,r,3) admit a partial generalization in the sense of the Erdos-Ginzburg-Ziv theorem. This work begins the off-diagonal case of the results of Bialostocki, Erdos, and Lefmann.