Two results on cardinal invariants at uncountable cardinals

الأصل · EN

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 κ, b(κ) = κ⁺ implies a(κ) = κ⁺. This improves an earlier result of Blass, Hyttinen, and Zhang. It is also shown that if κ≥ ω is an uncountable regular cardinal, then d(κ) ≤ r(κ). This result partially dualizes an earlier theorem of the authors.