Extremal bounds for pattern avoidance in multidimensional 0-1 matrices
Publication Date
2-1-2025
Document Type
Article
Publication Title
Discrete Mathematics
Volume
348
Issue
2
DOI
10.1016/j.disc.2024.114303
Abstract
A 0-1 matrix M contains another 0-1 matrix P if some submatrix of M can be turned into P by changing any number of 1-entries to 0-entries. The 0-1 matrix M is P-saturated where P is a family of 0-1 matrices if M avoids every element of P and changing any 0-entry of M to a 1-entry introduces a copy of some element of P. The extremal function ex(n,P) and saturation function sat(n,P) are the maximum and minimum possible number of 1-entries in an n×n P-saturated 0-1 matrix, respectively, and the semisaturation function ssat(n,P) is the minimum possible number of 1-entries in an n×n P-semisaturated 0-1 matrix M, i.e., changing any 0-entry in M to a 1-entry introduces a new copy of some element of P. We study these functions of multidimensional 0-1 matrices. In particular, we give upper bounds on parameters of minimally non-O(nd−1) d-dimensional 0-1 matrices, generalized from minimally nonlinear 0-1 matrices in two dimensions, and we show the existence of infinitely many minimally non-O(nd−1) d-dimensional 0-1 matrices with all dimensions of length greater than 1. For any positive integers k,d and integer r∈[0,d−1], we construct a family of d-dimensional 0-1 matrices with both extremal function and saturation function exactly knr for sufficiently large n. We show that no family of d-dimensional 0-1 matrices has saturation function strictly between O(1) and Θ(n) and we construct a family of d-dimensional 0-1 matrices with bounded saturation function and extremal function Ω(nd−ϵ) for any ϵ>0. Up to a constant multiplicative factor, we fully settle the problem of characterizing the semisaturation function of families of d-dimensional 0-1 matrices, which we prove to always be Θ(nr) for some integer r∈[0,d−1].
Funding Number
111-2115-M-008-010-MY2
Funding Sponsor
Ministry of Science and Technology, Taiwan
Keywords
0-1 matrix, Extremal function, Multidimensional matrix, Pattern avoidance, Saturation function
Department
Mathematics and Statistics
Recommended Citation
Jesse Geneson and Shen Fu Tsai. "Extremal bounds for pattern avoidance in multidimensional 0-1 matrices" Discrete Mathematics (2025). https://doi.org/10.1016/j.disc.2024.114303
