An exact characterization of saturation for permutation matrices

A 0-1 matrix \(M\) contains a pattern \(P\) if we can obtain from by deleting rows and/or columns and turning arbitrary 1-entries into 0s. The saturation function \(\mathrm{sat}(P,n)\) for indicates the minimum number of 1s in an \(n \times n\) that does not contain \(P\), but changing any 0-entry 1-entry creates occurrence \(P\). Fulek Keszegh recently showed each has either \(\mathcal{O}(1)\) or \(\Theta(n)\). We fully classify functions permutation matrices.Mathematics Subject Classifications: 05D99Keywords: Forbidden submatrices,