Consider a matrix $M$ chosen uniformly at random from a class of $m \times n$ matrices of zeros and ones with prescribed row and column sums. A partially filled matrix $D$ is a $\mathit{defining}$ $\mathit{set}$ for $M$ if $M$ is the unique member of its class that contains the entries in $D$. The $\mathit{size}$ of a defining set is the number of filled entries. A $\mathit{critical}$ $\mathit{set}$ is a defining set for which the removal of any entry stops it being a defining set. For some small fixed $\epsilon>0$, we assume that $n\le m=o(n^{1+\epsilon})$, and that $\lambda\le1/2$, where $\lambda$ is the proportion of entries of $M$ that equal $1$. We also assume that the row sums of $M$ do not vary by more than $\mathcal{O}(n^{1/2+\epsilon})$, and that the column sums do not vary by more than $\mathcal{O}(m^{1/2+\epsilon})$. Under these assumptions we show that $M$ almost surely has no defining set of size less than $\lambda mn-\mathcal{O}(m^{7/4+\epsilon})$. It follows that $M$ almost surely has no critical set of size more than $(1-\lambda)mn+\mathcal{O}(m^{7/4+\epsilon})$. Our results generalise a theorem of Cavenagh and Ramadurai, who examined the case when $\lambda=1/2$ and $n=m=2^k$ for an integer $k$.

中文翻译：

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大多数二元矩阵都有不小的定义集

考虑一个矩阵 $M$ 从一类 $m \times n$ 零矩阵和具有指定行和列总和的矩阵中随机均匀选择。部分填充的矩阵 $D$ 是 $M$ 的 $\mathit{defining}$ $\mathit{set}$，如果 $M$ 是包含 $D$ 中条目的类的唯一成员。定义集的 $\mathit{size}$ 是填充条目的数量。$\mathit{critical}$ $\mathit{set}$ 是一个定义集，删除任何条目都会使其成为定义集。对于一些小的固定 $\epsilon>0$，我们假设 $n\le m=o(n^{1+\epsilon})$，并且 $\lambda\le1/2$，其中 $\lambda$ 是等于 $1$ 的 $M$ 条目的比例。我们还假设 $M$ 的行总和的变化不超过 $\mathcal{O}(n^{1/2+\epsilon})$，并且列总和的变化不超过 $\ mathcal{O}(m^{1/2+\epsilon})$。在这些假设下，我们证明 $M$ 几乎肯定没有小于 $\lambda mn-\mathcal{O}(m^{7/4+\epsilon})$ 的定义集合。因此，$M$ 几乎肯定没有超过 $(1-\lambda)mn+\mathcal{O}(m^{7/4+\epsilon})$ 的临界大小集。我们的结果概括了 Cavenagh 和 Ramadurai 的定理，他们检查了当 $\lambda=1/2$ 和 $n=m=2^k$ 的情况下的整数 $k$。