Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: $\alpha$-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.

22
The categories of this paper are: chao-dyn
23
The categories of this paper are: solv-int
30
The categories of this paper are: q-alg
8
The categories of this paper are: hep-ex
9
The categories of this paper are: nlin
19
The categories of this paper are: eess
15
The categories of this paper are: nucl-ex
18
The categories of this paper are: econ
17
The categories of this paper are: q-fin
1
The categories of this paper are: cs
27
The categories of this paper are: comp-gas
26
The categories of this paper are: bayes-an
7
The categories of this paper are: hep-th
0
The categories of this paper are: hep-ph
28
The categories of this paper are: alg-geom
12
The categories of this paper are: nucl-th