Determine the Erdős–Kleitman extremal set-system quantity

Determine the maximum possible size of a family of subsets of {1,2,…,k} such that all pairwise unions are distinct, equivalently determining the maximum possible clique number of a graph with edge metric dimension at most k.

Background

The paper proves an exact equivalence between the maximum clique number among graphs with edge metric dimension at most k and an extremal set-theory problem posed by Erdős and Kleitman. The set-system problem concerns families whose pairwise unions are all distinct and is explicitly identified as open.