{"paper":{"title":"The linear refinement number and selection theory","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.LO"],"primary_cat":"math.GN","authors_text":"Boaz Tsaban, Micha{\\l} Machura, Saharon Shelah","submitted_at":"2014-04-08T18:03:36Z","abstract_excerpt":"The \\emph{linear refinement number} $\\mathfrak{lr}$ is the minimal cardinality of a centered family in $[\\omega]^\\omega$ such that no linearly ordered set in $([\\omega]^\\omega,\\subseteq^*)$ refines this family. The \\emph{linear excluded middle number} $\\mathfrak{lx}$ is a variation of $\\mathfrak{lr}$. We show that these numbers estimate the critical cardinalities of a number of selective covering properties. We compare these numbers to the classic combinatorial cardinal characteristics of the continuum. We prove that $\\mathfrak{lr}=\\mathfrak{lx}=\\mathfrak{fd}$ in all models where the continuum"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1404.2239","kind":"arxiv","version":1},"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"}