arXiv 1908.012672019-08math.CO

大多数二元矩阵没有小定义集

Most binary matrices have no small defining set

Carly Bodkin, Anita Liebenau, Ian M. Wanless

发表信息：: Disc. Math. 343 (2020), 112035
DOI：: 10.1016/j.disc.2020.112035

摘要：

考虑一个矩阵$M$，它是从一类由0和1组成的$m \times n$矩阵中随机均匀选取的，具有规定的行和列和。部分填充矩阵$D$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$\mathit{size}$是已填充条目的数量。$\mathit{critical}$ $\mathit{set}$是一个定义集，删除任何条目都会使其不再是定义集。对于一些小的固定的$ε>0$，我们假设$n\le m=o(n^{1+ε})$和$λ\le1/2$，其中$λ$是等于$1$的$M$的条目的比例。我们还假设$M$的行和变化不超过$\mathcal{O}(n^{1/2+ε})$，列和变化不超过$\mathcal{O}(m^{1/2+ε})$。在这些假设下，我们表明，$M$几乎肯定没有定义集的大小小于$λmn-\mathcal{O}(m^{7/4+ε})$。因此，$M$几乎肯定没有大于$(1-λ)mn+\mathcal{O}(m^{7/4+ε})$的临界集。我们的研究结果概括了一个定理的Cavenagh和Ramadurai，谁检查的情况下，当$λ=1/2$和$n=m=2^k$为整数$k$。

英文摘要：

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 $ε>0$, we assume that $n\le m=o(n^{1+ε})$, and that $λ\le1/2$, where $λ$ 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 $\mathcal{O}(n^{1/2+ε})$, and that the column sums do not vary by more than $\mathcal{O}(m^{1/2+ε})$. Under these assumptions we show that $M$ almost surely has no defining set of size less than $λmn-\mathcal{O}(m^{7/4+ε})$. It follows that $M$ almost surely has no critical set of size more than $(1-λ)mn+\mathcal{O}(m^{7/4+ε})$. Our results generalise a theorem of Cavenagh and Ramadurai, who examined the case when $λ=1/2$ and $n=m=2^k$ for an integer $k$.

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