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The equivalence of the Szemerédi and Petruska conjecture and the maximum order of $3$-uniform $τ$-critical hypergraphs

Recently we asymptotically resolved the long-standing Szemerédi and Petruska conjecture. Several decades ago Gyárfás 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 observed, via a straightforward but unpublished argument, that this conjecture is equivalent to the problem of determining the maximum order of a $3$-uniform $τ$-critical hypergraph. Consequently, an asymptotically tight upper bound for the maximum order of a $3$-uniform $τ$-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 Szemerédi and Petruska conjecture.