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

TL;DR

This paper proves the equivalence between the Szemerédi and Petruska conjecture and the maximum order problem of 3-uniform τ-critical hypergraphs, providing insights into longstanding combinatorial conjectures and their geometric applications.

Contribution

The paper offers a simple proof of the equivalence between the conjecture and hypergraph maximum order problem, and discusses related open problems and applications.

Findings

Asymptotic resolution of the Szemerédi and Petruska conjecture.

Establishment of the equivalence with the maximum order of 3-uniform τ-critical hypergraphs.

Discussion of related open problems and geometric applications.

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 observed, via a straightforward but unpublished argument, that this conjecture is equivalent to the problem of determining the maximum order of a $3$-uniform $\tau$-critical hypergraph.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