Let $X$ be a set, let $\mathcal{C}$ be a collection of subsets of $X$, and let $x_1, \dots , x_k \in X$. Say that the sequence $\{x_i\}_{i=1\dots k}$ is a *sequential transversal (of length $k$) through $\mathcal{C}$* if there exist $S_i \in \mathcal{C}$ for which $x_i\in S_i$ but $x_i \notin S_j$ if $j\lt i$. Say that $\{x_i\}$ is a *sequential antitransversal (of length $k$) through $\mathcal{C}$* if there exist $S_i \in \mathcal{C}$ for which $x_i\notin S_i$ but $x_i \in S_j$ if $j\lt i$, that is, if $\{x_i\}_{i=1\dots k}$ is a sequential transversal of length $k$ through the collection of complements of elements of $\mathcal{C}$.

Are there known conditions on $\mathcal{C}$ or the elements of $\mathcal{C}$ that guarantee the existence of a long sequential transversal, or that guarantee the existence of either a sequential transversal or a sequential antitransversal of some minimum length?

I found the term sequential transversal here. Is there another, more common, name for it in the literature, or a not-too-general general setting for which this is a specific case? A number of similar properties seem to be well-studied in combinatorics and order theory, but not this exact one.