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 \(\epsilon>0\), 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 \(k\).