Complexity of Extremal Set Decision Problem

Let a set \([n] = \{1,2,\ldots,n\}\) be given. Finding a subset \( S \) of \( 2^{[n]} \) with minimum cardinality such that, for any two distinct elements \( x, y \in [n] \), there exist disjoint subsets \( A_x, A_y \in \mathcal{S} \) such that \( x \in A_x \) and \( y \in A_y \) is called the \emph{extremal set} problem. In this paper, we define the Extremal Set Decision (ESD) Problem and study its complexity.