Most binary matrices have no small defining set

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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