论文标题
Szemerédi和Petruska猜想的等效性以及$ 3 $均匀$τ$的最高顺序
The equivalence of the Szemerédi and Petruska conjecture and the maximum order of $3$-uniform $τ$-critical hypergraphs
论文作者
论文摘要
最近，我们渐近地解决了长期的Szemerédi和Petruska猜想。几十年前，Gyárfás等人。通过直接但未发表的论点观察到，这种猜想等同于确定$ 3 $ - 均匀$τ$ - 关键超图的最大顺序的问题。因此，渐近紧密的上限，以$ 3 $ - 均匀$τ$ - 关键的超图的最大顺序遵循我们最近的工作，这对这种等价性产生了兴趣。在这份同伴论文中，我们提供了简单的证据。我们还提出了相关的背景，并提到了Szemerédi和Petruska猜想的组合几何应用。
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 observed, via a straightforward but unpublished argument, that this conjecture is equivalent to the problem of determining the maximum order of a $3$-uniform $τ$-critical hypergraph.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.
