{"paper":{"title":"$\\alpha$-Large Families and Applications to Banach Space Theory","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.FA","authors_text":"Pavlos Motakis, Spiros A. Argyros","submitted_at":"2013-02-04T15:18:08Z","abstract_excerpt":"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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1302.0715","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}