For a cardinal $\kappa$ by $[\kappa]^{<\kappa}$ we denote the family of all subsets of cardinality $<\kappa$ in $\kappa$. 



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> **Question.** Assume that for an infinite cardinal $\kappa$ there exists a family $\mathcal B\subset[\kappa]^{<\kappa}$ of cardinality $|\mathcal B|=\kappa$ such that for any $A\in[\kappa]^{<\kappa}$ the set $\{B\in\mathcal B:A\not\subset B\}$ has cardinality $<\kappa$. Is $\kappa$ a regular cardinal?
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A possible characterization of regular cardinals?
Yes. The assumption implies that $\mathcal{B}$ is cofinal in the poset $[\kappa]^{<\kappa}$, and hence that this poset has cofinality $\le\kappa$. This implies that $\kappa$ is regular, [by this answer](https://mathoverflow.net/a/313447/14094).


