Saturation of 0-1 Matrices

TL;DR

This work advances the theory of pattern saturation for 0-1 matrices by establishing a sharp dichotomy: for any forbidden pattern $P$, the saturation function $sat(n,P)$ is either $\Theta(n)$ or $O(1)$. It delivers a thorough classification for all patterns with at most four ones, identifies multiple infinite families with bounded or linear saturation, and systematically studies how matrix operations (notably the Kronecker product) and row/column insertions affect saturation. A key methodological contribution is the introduction of witness graphs, plus a linear programming framework to compute $sat(m,n,P)$ for small instances, which supports conjectures about exact formulas for broader pattern families. The paper also extends the theory to fixed dimensions and multidimensional matrices, including the first example of a $d$-dimensional pattern with bounded saturation for $d>2$, and establishes general asymptotic behavior when fixing several dimensions. These results refine our understanding of when pattern avoidance in matrices can be kept small and illuminate structural tools for further classifications and computations.

Abstract

A 0-1 matrix $M$ contains a 0-1 matrix $P$ if $M$ has a submatrix $P'$ which can be turned into $P$ by changing some of the ones to zeroes. Matrix $M$ is $P$-saturated if $M$ does not contain $P$, but any matrix $M'$ derived from $M$ by changing a zero to a one must contain $P$. The saturation function $sat(n,P)$ is defined as the minimum number of ones of an $n \times n$ $P$-saturated 0-1 matrix. Fulek and Keszegh showed that each pattern $P$ has $sat(n,P) = O(1)$ or $sat(n,P) = Θ(n)$. This leads to the natural problem of classifying forbidden 0-1 matrices according to whether they have linear or bounded saturation functions. Some progress has been made on this problem: multiple infinite families of matrices with bounded saturation function and other families with linear saturation function have been identified. We answer this question for all patterns with at most four ones, as well as several specific patterns with more ones, including multiple new infinite families. We also consider the effects of certain matrix operations, including the Kronecker product and insertion of empty rows and columns. Additionally, we consider the simpler case of fixing one dimension, extending results of (Fulek and Keszegh, 2021) and (Berendsohn, 2021). We also generalize some results to $d$-dimensional saturation.

Figures (4)

• Figure 1: An example of $G_3(W)$ for pattern $P$ and witness $W$. This is isomorphic to the complete bipartite directed graph $K_{3,3}$.
• Figure 2: Finding patterns $S$ and $S'$ in candidate witnesses.
• Figure 3: Examples of $Q_2$ in $W_V(Q_2)$ and $W_H(Q_2)$
• Figure 4: Finding $P_0$ in an example for $Q_3$ as described in Proposition \ref{['prop:q3_like']}. The one entries in red are those in $P_{j_2}$, while those in blue are those in $P_{a_1}$. The entries marked by $\circ$ are those in $P_0$.

Theorems & Definitions (86)

• Definition 1.1: Saturation Function
• Proposition 2.1: BC
• Theorem 2.2: FK
• Claim 2.3: FK
• Definition 2.4: Semisaturation
• Theorem 2.5: FK
• Theorem 2.6: FK
• Definition 2.7: Witness
• Theorem 2.8: berendsohn21FK
• Theorem 2.9: berendsohn21