On three sets with nondecreasing diameter

Abstract

Let [a, b] denote the integers between a and b inclusive and, for a finite subset X ⊆ Z, let diam (X) = max(X) − min(X). We write X <p Y provided max(X) < min(Y ). For a positive integer m, let f (m, m, m; 2) be the least integer N such that any 2-coloring ∆ : [1, N ] → {0, 1} has three monochromatic m-sets B1, B2, B3 ⊆ [1, N ] (not necessarily of the same color) with B1 <p B2 <p B3 and diam (B1) ≤ diam (B2) ≤ diam (B3). Improving upon upper and lower bounds of Bialostocki, Erdős and Lefmann, we show that f (m, m, m; 2) = 8m − 5 + ⌊ 2m−2 3 ⌋ + δ for m ≥ 2, where δ = 1 if m ∈ {2, 5} and δ = 0 otherwise.

On Monochromatic Pairs of Nondecreasing Diameters

“…Recent work has been done on generalizations of Question 2, as well as investigations into its relationship to a theorem by Erdős, Ginzburg, and Ziv (see [3,4,6,7,8,10] for examples). With possibly the only exceptions being the bounds given in Theorem 4(d) and (f), as well as Theorem 5(b), all recent work in evaluating f (m, r, t) focuses on parameter families with t = 2 and more recently t = 3 [1]. In our work we compute values for f (2, 2, t) for arbitrarily large t which, as far as we can tell, is the first known set of actual values of f (m, r, t) with m, r 2 and arbitrarily large t, rather than asymptotic results.…”

“…• (2,1,2). In this case ∆ ends with baaab, meaning the triples were not correctly identified in its AST partition.…”