Monochromatic and Zero-Sum Sets of Nondecreasing Modified Diameter

Let m be a positive integer whose smallest prime divisor is denoted by p, and let Zm denote the cyclic group of residues modulo m. For a set B = {x1, x2, ..., xm} of m integers satisfying x1 < x2 < ... < xm, and an integer j satisfying 2 <= j <= m, define gj(B) = xj − x1. Furthermore, define fj(m, 2) (define fj(m, Zm)) to be the least integer N such that for every coloring Delta: {1, ..., N} -> {0, 1} (every coloring Delta : {1, ..., N} -> Zm), there exist two m-sets [see Abstract in the PDF].