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Small forbidden configurations. V: Exact bounds for $4 \times 2$ cases. (English)
Stud. Sci. Math. Hung. 48, No. 1, 1-22 (2011).
The matrices under consideration can be interpreted as incidence matrices of systems of subsets of a finite set; a $(0,1)$-matrix is simple if it has no repeated columns (i.e., if no two of the subsets under consideration coincide). For a fixed $k\times l$ $(0,1)$-matrix $F$ (the forbidden configuration), $\text{forb}(m,F)$ is the largest $n$ for which there exists an $m\times n$ simple $(0,1)$ matrix having no submatrix which is a row and/or column permutation of $F$. $F_{abcd}$ is the $(a+b+c+d) \times 2$ matrix consisting of $a$ rows of $\left[\matrix 1&1\endmatrix\right]$, $b$ rows of $\left[\matrix 1&0\endmatrix\right]$, $c$ rows of $\left[\matrix 0&1\endmatrix\right]$, and $d$ rows of $\left[\matrix 0&0\endmatrix\right]$. With the exception of $F_{2110}$ (considered in \S7), the paper computes $\text{forb}(m,F_{abcd})$ for all cases where $a+b+c+d=4$. The paper continues work begun in {\it R. P. Anstee, J. R. Griggs} and {\it A. Sali} [Graphs Comb. 13, 97‒118 (1997; Zbl 0878.05019)], {\it R. P. Anstee, R. Ferguson} and {\it A. Sali} [Electron. J. Comb. 8, Research paper R4, 24 p. (2001; Zbl 0960.05100)], {\it R. P. Anstee} and {\it N. Kamoosi} [Electron. J. Comb. 14, Research Paper R79, 33 p. (2007; Zbl 1184.05130)], {\it R. P. Anstee} and {\it A. Sali} [Combinatorica 25, 503‒518 (2005; Zbl 1107.05090)]. From the authors’ abstract: “ Our results suggest that determining $\text{forb}(m,F_{2110})$ is heavily related to designs, and we offer some constructions of matrices avoiding $F_{2110}$, using existing designs.
Reviewer: William G. Brown (Montreal)