Monochromatic and zero-sum sets of nondecreasing modified diameter

Abstract

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 Δ : {1,..., N} → {0,1} (every coloring Δ : {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) |δ(Bi)| = 1 for i = 1, 2 (and (iii) Σ x∈Bi δ(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 + m/p - 6. Moreover, we show that fj(m, 2) ≥ 4m - 2 + (j - 1)k, where k = [(-1 + √8m-9+j/j-i)/2] and, if m is prime or j ≥ m/p + p - 1, that f j(m, ℤm) ≤ 6m - 4. We conclude by showing f m-1(m, 2) = fm-1(m, ℤm) for m ≥ 9.

Publication Title

Electronic Journal of Combinatorics

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