Abstract
Motivated both by the work of Anstee, Griggs, and Sali on forbidden submatrices and also by the extremal sat-function for graphs, we introduce sat-type problems for matrices. Let \({\mathcal{F}}\) be a family of k-row matrices. A matrix M is called \({\mathcal{F}}\)-admissible if M contains no submatrix \({F \in \mathcal{F}}\) (as a row and column permutation of F). A matrix M without repeated columns is \({\mathcal{F}}\)-saturated if M is \({\mathcal{F}}\)-admissible but the addition of any column not present in M violates this property. In this paper we consider the function sat(\({n, \mathcal{F}}\)) which is the minimal number of columns of an \({\mathcal{F}}\)-saturated matrix with n rows. We establish the estimate sat\({(n, \mathcal{F})=O(n^{k-1})}\) for any family \({\mathcal{F}}\) of k-row matrices and also compute the sat-function for a few small forbidden matrices.
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Research of the second author was partially supported by the National Science Foundation, Grants DMS-0758057 and DMS-1100215.
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Dudek, A., Pikhurko, O. & Thomason, A. On Minimum Saturated Matrices. Graphs and Combinatorics 29, 1269–1286 (2013). https://doi.org/10.1007/s00373-012-1199-2
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DOI: https://doi.org/10.1007/s00373-012-1199-2