论文标题
部分可观测时空混沌系统的无模型预测
An asymptotic resolution of a conjecture of Szemerédi and Petruska
论文作者
论文摘要
考虑使用$ 3 $均匀的订单$ n $，其中包数字$ k $，以使其所有$ k $ cliques的交集是空的。 Szemerédi和Petruska证明了$ n \ leq 8M^2+3M $，对于固定$ M = N-K $，他们猜想了尖锐的界限$ n \ leq {m+2 \ select 2} $。已知此问题等同于确定$τ$ - 临界$ 3 $ 3 $ - 均匀的超图和横向数字$ m $（详细信息也可以在同伴论文中找到详细信息：arxiv：2204.02859）。 $ n \ leq \ frac {3} {4} m^2+m+1 $是使用$τ$ -Critical-Critical-Critical HyperGraphs获得的。在这里，我们提出了一种替代方法，这是Szemerédi和Petruska引入的迭代分解过程的组合，以及Bollobás的偏斜版本在设定对系统上的偏斜版本。新方法改善了绑定到$ n \ leq {m + 2 \ select 2} + o（m^{{5}/{3}}}}）$，从而渐近地解决猜想。
Consider a $3$-uniform hypergraph of order $n$ with clique number $k$ such that the intersection of all its $k$-cliques is empty. Szemerédi 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 $τ$-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 $τ$-critical hypergraphs. Here we propose an alternative approach, a combination of the iterative decomposition process introduced by Szemerédi and Petruska with the skew version of Bollobás'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.
