{"paper":{"title":"Two results on cardinal invariants at uncountable cardinals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Dilip Raghavan, Saharon Shelah","submitted_at":"2018-01-29T05:54:28Z","abstract_excerpt":"We prove two ZFC theorems about cardinal invariants above the continuum which are in sharp contrast to well-known facts about these same invariants at the continuum. It is shown that for an uncountable regular cardinal $\\kappa$, $\\mathfrak{b}(\\kappa) = {\\kappa}^{+}$ implies $\\mathfrak{a}(\\kappa) = {\\kappa}^{+}$. This improves an earlier result of Blass, Hyttinen, and Zhang. It is also shown that if $\\kappa \\geq {\\beth}_{\\omega}$ is an uncountable regular cardinal, then $\\mathfrak{d}(\\kappa) \\leq \\mathfrak{r}(\\kappa)$. This result partially dualizes an earlier theorem of the authors."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.09369","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"}