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arXiv · 2204.02859

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

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 observed, via a straightforward but unpublished argument, that this conjecture is equivalent to the problem of determining the maximum order of a $3$-uniform $\tau$-critical hypergraph.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

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André E. Kézdy, Jenő Lehel. 2022-04-06. The equivalence of the Szemer\'edi and Petruska conjecture and the maximum order of $3$-uniform $\tau$-critical hypergraphs. https://arxiv.org/abs/2204.02859

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