The following seems like a natural problem and I'm surprised I can't find any literature on it... but maybe it's because I don't know the name for it.
Given a list of sets $S_1, S_2, S_3, \ldots$ Can we make $k$ additions (that is, adding an element to any single $S_i$ ) so that every pair of sets has nonempty intersection?
If we treat each subset $S_i$ as a vertex and say two vertices are adjacent if their corresponding sets have at least one common element, this is like an intersection model for a graph and we are augmenting the representation to turn the graph into a clique. I think of this as some sort of "clique completion" problem, but I don't know if it has been studied in another context under a different name.
Does anyone know of this problem?