  Recently we asymptotically resolved the long-standing Szemer\'edi and
Petruska conjecture. Several decades ago Gy\'arf\'as et al. observed, via a
straightforward but unpublished argument, that this conjecture is equivalent to
the problem of determining the maximum order of a $3$-uniform $\tau$-critical
hypergraph. Consequently, an asymptotically tight upper bound for the maximum
order of a $3$-uniform $\tau$-critical hypergraph follows from our recent work,
reawakening interest in this equivalence. In this companion paper we supply a
simple proof of this equivalence. We also present related background with open
problems, and mention combinatorial geometry applications of the Szemer\'edi
and Petruska conjecture.
