Exact Bounds for Forbidden Configurations and the Extremal Matrices

TL;DR

The paper studies exact bounds for forbidding the configuration $F(0,p,1,0)$ in simple $(0,1)$-matrices via a graph-theoretic row framework. It develops a row-graph construction, the What is Missing principle, and a suite of lemmas (Deletion, Upper Bound, Transitivity) to constrain the possible non-constant columns and component structures. A central clique-components analysis shows that extremal matrices decompose into clique-based blocks, enabling precise bounds and complete extremal descriptions for key cases, notably $p=3$ and $p=6$, with tight constructions for several congruence classes of $m$. The methods yield exact or near-exact bounds for all $p\in\{3,4,5,6,7,8,9\}$ and provide a scalable construction scheme that could guide further extensions to larger $p$ in the forbidden-configuration paradigm.

Abstract

Let $F$ be a $k\times \ell$ (0,1)-matrix. A matrix is simple if it is a (0,1)-matrix with no repeated columns. A (0,1)-matrix $A$ is said to have a $F$ as a configuration if there is a submatrix of $A$ which is a row and column permutation of $F$. In the language of sets, a configuration is a trace. Let $\mathrm{Avoid}(m,F)$ be all simple $m$-rowed matrices $A$ with no configuration $F$. Define $\mathrm{forb}(m,F)$ as the maximum number of columns of any matrix in $\mathrm{Avoid}(m,F)$. The $2\times (p+1)$ (0,1)-matrix $F(0,p,1,0)$ consists of a row of $p$ 1's and a row of one 1 in the remaining column. The paper determines $\mathrm{forb}(m,F(0,p,1,0))$ for $1\le p\le 9$ and the extremal matrices are characterized. A construction may be extremal for all $p$.

Theorems & Definitions (16)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 2.1
• Conjecture 2.2
• Definition 3.1
• Lemma 3.2
• Lemma 3.3
• Lemma 3.4