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This is a follow-up question to an older question. Let $X\neq \emptyset$ be a set. We say that ${\cal C}\subseteq {\cal P}(X)$ is a cover if $\bigcup {\cal C} = X$, and we call ${\cal C}$ linear if $|...

Motivation. The starting point of this question is the trivial observation that if we cover $\mathbb{N}$ with $$\big\{\{0,\ldots n\}: n\in \mathbb{N}\big\},$$ then this cover doesn't have a minimal ...

Let $\kappa < \lambda < \mu$ be infinite cardinals. Is there a collection ${\cal U}\subseteq {\cal P}(\mu)$ of subsets of $\mu$ with the following properties? for all $U\in {\cal U}$ we have $|...