Ramsey-Sperner theory

Let [n] denote the n-set {1, 2,..., n}, let k, l ≥ 1 be integers. Define f_l(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}A_l+1 in F in which the set of added elements, A_l+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→^∞ f_l(n,k)∼φ{symbol}_l(k)⌊ 1 2n⌋ⁿ and φ{symbol}_l(k)∼ φk 4logk as k→^∞. Several problems remain open.