Intermingled ascending wave m-sets

摘要

Given a coloring of Z(+), we call a monochromatic set A = {a(l) < a(2) < ... < a(m)} an m-set. The diameter ofA is a(m)-a(l). 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 msets, done by Bialostocki, Erdos, 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 {a(l) < a(2) < ... < a(m)} and {b(1) < b(2) < ... < b(m)} with a(m) < b(1), such that b(1) - a(1) <= b(2) - a(2) <=...<= b(m) - a(m) is n(m) = 6m - 5. The r-coloring case is also investigated. (c) 2015 Elsevier B.V. All rights reserved.