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 observed, via astraightforward but unpublished argument, that this conjecture is equivalent tothe problem of determining the maximum order of a $3$-uniform $\tau$-criticalhypergraph.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

The equivalence of the Szemer\'edi and Petruska conjecture and the maximum order of $3$-uniform $\tau$-critical hypergraphs