$\alpha$-Large Families and Applications to Banach Space Theory

The notion of $\alpha$-large families of finite subsets of an infinite set is defined for every countable ordinal number $\alpha$, extending the known notion of large families. The definition of the $\alpha$-large families is based on the transfinite hierarchy of the Schreier families $\mathcal{S}_{\alpha}, \alpha<\omega_1$. 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 $\mathfrak{X}_{2^{\aleph_0}}^{\alpha}$, $\alpha<\omega_1$ with density the continuum, such that every bounded non norm convergent sequence $\{x_k\}_k$ has a subsequence generating $\ell_1^{\alpha}$ as a spreading model.