$α$-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 $\mathcal{S}_α, α<ω_1$. We prove the existence of such families on the cardinal number $2^{\aleph_0}$ and we study their properties. As an application, based on those families we construct a reflexive space $\mathfrak{X}_{2^{\aleph_0}}^α$, $α<ω_1$ with density the continuum, such that every bounded non norm convergent sequence $\{x_k\}_k$ has a subsequence generating $\ell_1^α$ as a spreading model.

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