Multivalued forbidden numbers of two-rowed configurations -- the missing cases

TL;DR

This work extends forbidden configurations to multivalued matrices with $s=3$, resolving missing $2$-row cases by establishing the asymptotic gap between $\mathrm{forb}(m,3,p\cdot K_2)$ and $\mathrm{forb}(m,3,p\cdot I_2)$ and by obtaining exact values for a broad class of $2$-row configurations $F(a,p,p,d)$. The core methods combine a standard matrix decomposition $A=B\mid C$, a transitivity lemma on an auxiliary row-pair graph, and precise counting of zero and one marks to control the structure of extremal matrices. The main findings include an asymptotic formula $g_p \sim \tfrac{1}{2} p(\log_2 p)^2$ and a taxonomy of exact regimes where the upper and lower bounds coincide, especially for small $p$ and in wide ranges for larger $p$ governed by $r=\lceil\log_2(p-1)\rceil$ and $N_r$. These results advance the understanding of exponential forbidden numbers in multivalued matrix configurations and lay groundwork for tackling the remaining missing cases.

Abstract

The present paper considers extremal combinatorics questions in the language of matrices. An $s$-matrix is a matrix with entries in $\{0,1,\ldots, s-1\}$. An $s$-matrix is simple if it has no repeated columns. A matrix $F$ is a configuration in a matrix $A$, denoted $F\prec A$, if it is a row/column permutation of a submatrix of $A$. $\text{Avoid}(m,s,F)$ is the set of $m$-rowed, simple $s$-matrices not containing a configuration of $F$ and $\text{forb}(m,s, F)=\max\{|A|\colon A \in \text{Avoid}(m,s,F)\}$. Dillon and Sali initiated the systematic study of $\text{forb}(m,s, F)$ for $2$-matrices $F$, and computed $\text{forb}(m,s, F)$ for all 2-rowed $F$ when $s>3$. In this paper we tackle the remaining cases when $s=3$. In particular, we determine the asymptotics of $\text{forb}(m,3,p\cdot K_2)-\text{forb}(m,3,p\cdot I_2)$ for $p>3$, where $K_2$ is the $2\times 4$ simple $2$-matrix and $I_2$ is the $2\times 2$ identity matrix, as well as the exact values of $\text{forb}(m,3,F)$ for many 2-rowed $2$-matrices $F$.

Theorems & Definitions (54)

• Theorem 1.1
• Theorem 1.2
• Lemma 1.3
• Proposition 1.4
• Theorem 1.5
• Theorem 1.6
• Corollary 1.7
• Proposition 1.8
• Proposition 1.9
• Definition 2.1