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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2507.19336 |
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Table of Contents:
- Let $F$ be a $k\times \ell$ (0,1)-matrix. Define a (0,1)-matrix $A$ to have a $F$ as a \emph{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 \emph{trace}. Define a matrix to be {\it simple} if it is a (0,1)-matrix with no repeated columns. 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 $\mathrm{Avoid}(m,F)$. Determining $\mathrm{forb}(m,F)$ requires determining bounds and constructions of matrices in $\mathrm{Avoid}(m,F)$. The paper considers some column maximal $k$-rowed simple $F$ that have the bound $Θ(m^{k-2})$ and yet adding a column increases bound to $Ω(m^{k-1})$. By a construction, $\mathrm{forb(m,F)}$ is determined exactly.