On Minimum Saturated Matrices

TL;DR

This paper introduces and analyzes the sat-function for matrices avoiding certain submatrices, establishing bounds and exact values for specific cases, inspired by extremal graph theory.

Contribution

It defines the sat-function for forbidden submatrix problems and proves an upper bound of O(n^{k-1}) for any family of k-row matrices, including exact computations for small cases.

Findings

sat(n,F)=O(n^{k-1}) for any family F of k-row matrices

Computed sat-function for specific small forbidden matrices

Established bounds inspired by extremal graph theory

Abstract

Motivated by the work of Anstee, Griggs, and Sali on forbidden submatrices and the extremal sat-function for graphs, we introduce sat-type problems for matrices. Let F be a family of k-row matrices. A matrix M is called F-admissible if M contains no submatrix G\in F (as a row and column permutation of G). A matrix M without repeated columns is F-saturated if M is F-admissible but the addition of any column not present in M violates this property. In this paper we consider the function sat(n,F) which is the minimum number of columns of an F-saturated matrix with n rows. We establish the estimate sat(n,F)=O(n^{k-1}) for any family F of k-row matrices and also compute the sat-function for a few small forbidden matrices.