Intermingled ascending wave m-sets

Given a coloring of Z+, we call a monochromatic set A = {a1 < a2 < · · · < am} an m-set. The diameter of A is am a1. Given two m-sets A and B, we say that they are non-overlapping if max(A) < min(B) or max(B) < min(A). The original study of non-overlapping m-sets, done by Bialostocki, Erdős, and Lefmann, concerned non-decreasing diameters. We investigate an “intermingling” of certain subset diameters of non-overlapping m-sets. In particular, we show that, for every integer m 2, the minimum integer n(m) such that every 2-coloring of [1, n(m)] admits two m-sets {a1 < a2 < · · · < am} and {b1 < b2 < · · · < bm} with am < b1, such that b1 a1  b2 a2  · · ·  bm am is n(m) = 6m 5. The r-coloring case is also investigated.