α-Large families and applications to Banach space theory

The notion of α-large families of finite subsets of an infinite set is defined for every countable ordinal number α, extending the known notion of large families. The definition of the α  -large families is based on the transfinite hierarchy of the Schreier families SαSα, α<ω1α<ω1. We prove the existence of such families on the cardinal number 2ℵ02ℵ0 and we study their properties. As an application, based on those families we construct a reflexive space X2ℵ0α, α<ω1α<ω1 with density the continuum, such that every bounded non-norm convergent sequence {xk}k{xk}k has a subsequence generating ℓ1α as a spreading model.