On three sets with nondecreasing diameter

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.