On Monochromatic Sets of Integers Whose Diameters Form a Monotone Sequence

Let g(m, t) denote the minimum integer s such that for every 2-coloring of the integers in the interval [1, s], there exist t subsets A1, A2, . . . , At, of size m satisfying: (i) Ai for every i = 1, 2, . . . , t is monochromatic (not necessarily the same color) (ii) max(Ai) ≤ min(Ai+1) for every i = 1, 2, . . . , t − 1, and (iii) either diam(Ai) ≤ diam(Ai+1) for every i = 1, 2, . . . , t − 1 or diam(Ai) ≥ diam(Ai+1) for every i = 1, 2, . . . , t− 1. We prove that 2(m− 1)(t + 1) + 1 ≤ g(m, t) ≤ [(t− 1)2 + 1](2m− 1) for every integer m and t, where m ≥ 2 and t ≥ 3. Furthermore, we determine that g(m, 3) = 8m− 5.