论文标题
指数多价禁止配置
Exponential multivalued forbidden configurations
论文作者
论文摘要
禁止数字$ \ mathrm {forb}（m，f）$表示，它在$ m $ $ $（0,1）$ - 矩阵中的最大数量唯一的列数，而没有$ f $的行和列置换列的子矩阵，它已在极端集合理论中得到了广泛研究。最近，此功能扩展到$ r $ - matrices，其条目位于$ \ {0,1，\ dots，r-1 \} $中。广义禁止数字的组合学不那么精心研究。在本文中，我们提供了许多$（0,1）$ - 矩阵$ f $的确切界限，包括$ r> 3 $时的所有$ 2 $ rowed矩阵。我们还证明了$ 2 \ times 2 $ Identity矩阵的稳定性结果。在此过程中，我们在案例$ r = 2 $，$ r = 3 $和$ r> 3 $之间暴露了一些有趣的定性差异。
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 $r > 3$.
