10.46298/dmtcs.6613
Dillon, Travis
Travis
Dillon
Sali, Attila
Attila
Sali
Exponential multivalued forbidden configurations
episciences.org
2021
Mathematics - Combinatorics
contact@episciences.org
episciences.org
2020-07-02T22:14:46+02:00
2021-08-23T23:08:42+02:00
2021-03-23
eng
Journal article
https://dmtcs.episciences.org/6613
arXiv:2006.16305
1365-8050
PDF
1
Discrete Mathematics & Theoretical Computer Science ; vol. 23 no. 1 ; Combinatorics ; 1365-8050
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 $r > 3$.
Comment: 12 pages; v3: formatted for DMTCS; v2: Corollary 3.2 added, typos
fixed, some proofs clarified