Let $\mathcal B=\mathcal B_{k,n,p}$ be a random collection of $k$-subsets of $[n]$ where each possible set is present independently with probability $p$. Let $\cal E_{\mathcal B}$ be the event that $\mathcal B$ defines the set of bases of a matroid. We prove that If $p= 1-\frac{c_n}{(k(n-k)\binom nk)^{1/2}}$ where $0\leq c_n\leq \infty$, then \[ \lim_{n\to\infty}\Pr[\cal E_{\cal B}\mid |\cal B|\geq2]=\begin{cases}1&c_n\to0.\\e^{-c^2/2}&c_n\to c.\\0&c_n\to \infty.\end{cases}\] In addition, we identify a condition preventing the occurence of $\cal E_{\cal B}$ and prove a hitting time version for the occurence of $\cal B$. We also prove that when $\cal E_{\mathcal B}$ occurs, $\mathcal B$ defines a sparse paving matroid w.h.p. In addition, study a greedy algorithm that produces a random matroid defined by a collection of hyperplanes. We use this to improve the estimates in \cite{HPV} on $\log m(n,k),\log p(n,k), \log s(n,k)$ where $ m(n, k), p(n, k), s(n, k)$ denote the number of matroids, paving matroids, and sparse paving matroids (respectively) of rank $k$ on $[n]$. Our improvement lies in that we can deal with $k$ growing slowly with $n$ as opposed to $k=O(1)$ in \cite{HPV}. More generally, we obtain estimates for the number of matchings in nearly-regular hypergraphs with small codegree, which may be of independent interest.

翻译：设 $\mathcal B=\mathcal B_{k,n,p}$ 为 $[n]$ 所有 $k$-子集的随机集合，其中每个子集以独立概率 $p$ 出现。令 $\cal E_{\mathcal B}$ 表示事件 $\mathcal B$ 定义了一个拟阵的基集。我们证明：若 $p= 1-\frac{c_n}{(k(n-k)\binom nk)^{1/2}}$，其中 $0\leq c_n\leq \infty$，则 \[ \lim_{n\to\infty}\Pr[\cal E_{\cal B}\mid |\cal B|\geq2]=\begin{cases}1&c_n\to0.\\e^{-c^2/2}&c_n\to c.\\0&c_n\to \infty.\end{cases}\] 此外，我们识别了阻止 $\cal E_{\cal B}$ 发生的条件，并证明了 $\cal B$ 发生的击时版本。我们还证明，当 $\cal E_{\mathcal B}$ 发生时，$\mathcal B$ 以高概率定义了一个稀疏铺砌拟阵。进一步，我们研究了一种由超平面集合定义随机拟阵的贪心算法，并用其改进 \cite{HPV} 中关于 $\log m(n,k),\log p(n,k), \log s(n,k)$ 的估计，其中 $m(n, k), p(n, k), s(n, k)$ 分别表示 $[n]$ 上秩为 $k$ 的拟阵、铺砌拟阵和稀疏铺砌拟阵的个数。我们的改进在于能够处理 $k$ 随 $n$ 缓慢增长的情形，而 \cite{HPV} 中仅适用于 $k=O(1)$。更一般地，我们获得了小余度近正则超图中匹配数的估计，该结果可能具有独立意义。