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 S α , α < ω 1 . We prove the existence of such families on the cardinal number 2 ℵ 0 and we study their properties. As an application, based on those families we construct a reflexive space X 2 ℵ 0 α , α < ω 1 with density the continuum, such that every bounded non-norm convergent sequence { x k } k has a subsequence generating ℓ 1 α as a spreading model.