Dillon, Travis and Sali, Attila - Exponential multivalued forbidden configurations

dmtcs:6613 - Discrete Mathematics & Theoretical Computer Science, March 23, 2021, vol. 23 no. 1
Exponential multivalued forbidden configurations

Authors: Dillon, Travis and Sali, Attila

The forbidden number $\mathrm{forb}(m,F)$, which denotes the maximum number of unique columns in an $m$-rowed $(0,1)$-matrix with no submatrix that is a row and column permutation of $F$, has been widely studied in extremal set theory. Recently, this function was extended to $r$-matrices, whose entries lie in $\{0,1,\dots,r-1\}$. The combinatorics of the generalized forbidden number is less well-studied. In this paper, we provide exact bounds for many $(0,1)$-matrices $F$, including all $2$-rowed matrices when $r > 3$. We also prove a stability result for the $2\times 2$ identity matrix. Along the way, we expose some interesting qualitative differences between the cases $r=2$, $r = 3$, and $r > 3$.

Volume: vol. 23 no. 1
Section: Combinatorics
Published on: March 23, 2021
Submitted on: July 2, 2020
Keywords: Mathematics - Combinatorics


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