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 ∆ : {1, . . . ,N} → {0, 1} (every coloring ∆ : {1, . . . ,N} → Zm), there exist two m-sets B1, B2 ⊂ {1, . . . ,N} satisfying: (i) max(B1) < min(B2), (ii) gj(B1) ≤ gj(B2), and (iii) |∆(Bi)| = 1 for i = 1, 2 (and (iii) ∑ x∈Bi ∆(x) = 0 for i = 1, 2). We prove that fj(m, 2) ≤ 5m − 3 for all j, with equality holding for j = m, and that fj(m, Zm) ≤ 8m + mp − 6. Moreover, we show that fj(m, 2) ≥ 4m − 2 + (j − 1)k, where k = ⌊( −1 + √ 8m−9+j j−1 ) /2 ⌋ , and, if m is prime or j ≥ mp + p − 1, that fj(m, Zm) ≤ 6m − 4. We conclude by showing fm−1(m, 2) = fm−1(m, Zm) for m ≥ 9.