On Minimum Saturated Matrices
Andrzej Dudek, Oleg Pikhurko, Andrew Thomason

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.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsLimits and Structures in Graph Theory · Graph theory and applications · Advanced Graph Theory Research
