Kerodon

Example 9.1.7.7. Let $\kappa $ be a regular cardinal with successor $\kappa ^{+}$. Then $\kappa \triangleleft \kappa ^{+}$. That is, if $S$ is a set of cardinality $\leq \kappa $, then there is a collection of subsets $\{ S_ i \} _{i \in I}$ having cardinality $\leq \kappa $ such that every $\kappa $-small subset of $S$ is contained in some $S_ i$. Without loss of generality, we may assume that $S = \mathrm{Ord}_{< \kappa }$ is the set of ordinals $\alpha $ satisfying $\alpha < \kappa $ (Remark 4.7.2.6). In this case, we can take $\{ S_ i \} _{i \in I}$ to be the collection of all subsets of the form $\mathrm{Ord}_{< \beta }$, where $\beta $ ranges over $\mathrm{Ord}_{< \kappa }$.