Significant work has been done on computing the ``average'' optimal solution value for various $\mathsf{NP}$-complete problems using the Erd\"{o}s-R\'{e}nyi model to establish \emph{critical thresholds}. Critical thresholds define narrow bounds for the optimal solution of a problem instance such that the probability that the solution value lies outside these bounds vanishes as the instance size approaches infinity. In this paper, we extend the Erd\"{o}s-R\'{e}nyi model to general hypergraphs on $n$ vertices and $M$ hyperedges. We consider the problem of determining critical thresholds for the largest cardinality matching, and we show that for $M=o(1.155^n)$ the size of the maximum cardinality matching is almost surely 1. On the other hand, if $M=\Theta(2^n)$ then the size of the maximum cardinality matching is $\Omega(n^{\frac12-\gamma})$ for an arbitrary $\gamma >0$. Lastly, we address the gap where $\Omega(1.155^n)=M=o(2^n)$ empirically through computer simulations.

翻译：利用Erdős-Rényi模型计算各类$\mathsf{NP}$完全问题"平均"最优解值以建立临界阈值的研究已取得显著进展。临界阈值定义了问题实例最优解的狭窄边界，使得当实例规模趋于无穷时，解值落在此边界之外的概率趋近于零。本文将Erdős-Rényi模型推广至具有$n$个顶点和$M$条超边的一般超图。我们研究了最大基数匹配临界阈值的确定问题，并证明当$M=o(1.155^n)$时，最大基数匹配的规模几乎必然为1。另一方面，若$M=\Theta(2^n)$，则对于任意$\gamma >0$，最大基数匹配的规模为$\Omega(n^{\frac12-\gamma})$。最后，我们通过计算机模拟实证研究了$\Omega(1.155^n)=M=o(2^n)$这一区间的情况。