Forbidden Configurations and Product Constructions

Abstract

Abstract.A simple matrix is a (0,1)-matrix with no repeated columns. For a (0,1)-matrix F , we define that a (0,1)-matrix A has F as a configuration if there is a submatrix of A which is a row and column permutation of F (trace is the set system version of a configuration). Let A denote the number of columns of A. We define forb(m, F ) = max{ A : A is m-rowed simple matrix and has no configuration F }. We extend this to a family F = {F 1 , F 2 , . . . , F t } and define forb(m, F) = max{ A : A is m-rowed simple matrix and has no configuration F ∈ F}. We consider products of matrices. Given an m 1 × n 1 matrix A and an m 2 × n 2 matrix B, we define the product A × B as the (m 1 + m 2 ) × n 1 n 2 matrix columns consist of all possible combinations obtained from placing a column of A on top of a column of B. Let I k denote the k × k identity matrix, let I c k denote the (0,1)-complement of I k and let T k denote the k × k upper triangular (0,1)-matrix with a 1 in position i, j if and only if i ≤ j. We show forb(m, {I 2 × I 2 , T 2 × T 2 }) is Θ(m 3/2 ) while obtaining a linear bound when forbidding all 2-fold products of all 2 × 2 (0,1)-simple matrices. For two matrices F, P , where P is m-rowed, let f (F, P ) = max A { A : A is m-rowed submatrix of P with no configuration F }. We establish f (I 2 × I 2 , I m/2 × I m/2 ) is Θ(m 3/2 ) whereas f (I 2 × T 2 , I m/2 × T m/2 ) and f (T 2 × T 2 , T m/2 × T m/2 ) are both Θ(m). Additional results are obtained. We use the results on patterns due to Marcus and Tardos and generalizations due to Klazar and Marcus, Balogh, Bollobás and Morris.

Forbidden Families of Minimal Quadratic and Cubic Configurations

“…The best current estimate for BB(k) is due to Anstee and Lu [8], BB(k) ≤ 2 ck 2 where c is absolute constant, independent of k. It could be tempting to extend Conjecture 1.1 to the case of forbidden families, as well. However, as it was shown in [5] forb(m, {I 2 × I 2 , T 2 × T 2 }) is Θ(m 3/2 ) despite the only products missing both I 2 × I 2 and T 2 × T 2 .are one-fold products. An even stronger observation is made in Remark 5.…”

Forbidden Families of Configurations

“…We use s = t−1 and then for any row r ∈ [m], obtain the decomposition (2). We wish to use (4). With F = {t • I k , F 2 (1, t, t, 1)} and s = t − 1, we have…”

“…You might note that I 2 × I 2 is the same as C 4 after a row and column permutation. Theorem 1.9 [4] We have that forb(m,…”

Forbidden Berge Hypergraphs

“…Sometimes the product may contain the best construction using the following idea from [5], that when given two matrices F, P where P is m-rowed then…”