They do not specify what is meant by $3$-critical. One definition in the literature is: a hypergraph is $3$-critical if there is a set of $3$ vertices which intersects every edge, but no such set of size $2$, and yet for any edge $e$ there is a pair of vertices which intersects every edge except $e$. Raphael Steiner observes that a $3$-critical hypergraph in this sense has bounded size, so this problem would be a finite computation, and perhaps is not what they meant.

An alternative definition is that a hypergraph is $3$-critical if it has chromatic number $3$, but its chromatic number becomes $2$ after deleting any edge or vertex.