## Abstract

Let [n] denote the n-set {1, 2,..., n}, let k, l ≥ 1 be integers. Define fl(n, k) as the minimum number f such that for every family F ⊆ 2[n] with {divides}F{divides}>f, for every k-coloring of [n], there exists a chain A1{subset not double equals}···{subset not double equals}Al+1 in F in which the set of added elements, Al+1-A1, is monochromatic. We survey the known results for l = 1. Applying them we prove for any fixed l that there exists a constant φ{symbol}l(k) such that as n→∞ fl(n,k)∼φ{symbol}l(k)⌊ 1 2n⌋n and φ{symbol}l(k)∼ φk 4logk as k→∞. Several problems remain open. © 1987.