Maximization problem on finite collection of finite sets
This problem is NP-complete. Let's reformulate it first: we have a bipartite graph, where

The left side corresponds to elements
The right side corresponds to sets
The edge $(u,v)$ means that $u \in v$.

Our goal is to find the bipartite clique with the maximum number of edges. As stated in Rene Peeters, &quot;The maximum edge biclique problem is NP-complete&quot;, the decision problem is NP-complete.