Pre-compact families of finite sets of integers and weakly null sequences in Banach spaces

In this paper we use the Nash-Williams theory of fronts and barriers to study weakly null sequences in Banach spaces. Specifically, we show how barriers relate to the classical fact that C(K) with K a countable compactum is c0-saturated. Another result relates the notion of a barrier to the Maurey–Rosenthal example of a weakly null sequence with no unconditional subsequences. In particular, we construct examples of weakly-null sequences which are α-unconditional but not β-unconditional.