szemersamodi和Petruska猜想的渐近解

Andr'e E. K'ezdy, JenHo Lehel
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引用次数: 3

摘要

考虑一个阶为$n$、团号为$k$的$3$ -一致超图,其所有$k$ -团的交集为空。szemersamudi和Petruska证明了$n\leq 8m^2+3m$,对于固定的$m=n-k$,他们推测了锐利界$n \leq {m+2 \choose 2}$。已知这个问题等价于确定一个截数为$m$的$\tau$ -临界$3$ -一致超图的最大阶(细节也可以在另一篇论文中找到:arXiv:2204.02859)。最著名的界$n\leq \frac{3}{4}m^2+m+1$是Tuza利用$\tau$临界超图的机制得到的。在这里,我们提出了一种替代方法,即szemer和Petruska引入的迭代分解过程与集对系统上Bollobás定理的扭曲版本的结合。新方法将界改进为$n\leq {m+2 \choose 2} + O(m^{{5}/{3}})$,渐近地解决了猜想。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
An asymptotic resolution of a conjecture of Szemerédi and Petruska
Consider a $3$-uniform hypergraph of order $n$ with clique number $k$ such that the intersection of all its $k$-cliques is empty. Szemer\'edi and Petruska proved $n\leq 8m^2+3m$, for fixed $m=n-k$, and they conjectured the sharp bound $n \leq {m+2 \choose 2}$. This problem is known to be equivalent to determining the maximum order of a $\tau$-critical $3$-uniform hypergraph with transversal number $m$ (details may also be found in a companion paper: arXiv:2204.02859). The best known bound, $n\leq \frac{3}{4}m^2+m+1$, was obtained by Tuza using the machinery of $\tau$-critical hypergraphs. Here we propose an alternative approach, a combination of the iterative decomposition process introduced by Szemer\'edi and Petruska with the skew version of Bollob\'as's theorem on set pair systems. The new approach improves the bound to $n\leq {m+2 \choose 2} + O(m^{{5}/{3}})$, resolving the conjecture asymptotically.
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