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

Abstract

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 observed, via a straightforward but unpublished argument, that this conjecture is equivalent to the problem of determining the maximum order of a $3$-uniform $\tau$-critical hypergraph.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.