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| Natura: | Preprint |
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2025
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| Accesso online: | https://arxiv.org/abs/2507.19336 |
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| _version_ | 1866909705328132096 |
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| author | Anstee, Richard P. Edens, Oakley Sahami, Arvin Sali, Attila |
| author_facet | Anstee, Richard P. Edens, Oakley Sahami, Arvin Sali, Attila |
| 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_19336 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Forbidden Configurations and Boundary Cases Anstee, Richard P. Edens, Oakley Sahami, Arvin Sali, Attila Combinatorics 05D05 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. |
| title | Forbidden Configurations and Boundary Cases |
| topic | Combinatorics 05D05 |
| url | https://arxiv.org/abs/2507.19336 |