Two refinements of the bound of Sauer, Perles and Shelah, and of Vapnik and Chervonenkis

Fleming

2011

Combinatorica

Self Cite

We say a matrix is simple if it is a (0,1)-matrix with no repeated columns. Given m and a k×l (0,1)-matrix F we define forb(m, F ) as the maximum number of columns in a simple m-rowed matrix A for which no k × l submatrix of A is a row and column permutation of F . In set theory notation, F is a forbidden trace. For all k-rowed F (simple or nonsimple) Füredi has shown that forb(m, F ) is O(m k ). We are able to determine for which k-rowed F we have that forb(m, F ) is O(m k−1 ) and for which k-rowed F we have that forb(m, F ) is Θ(m k ).We need a bound for a particular choice of F . Define D12 to be the k ×(2 k −2 k−2 −1) (0,1)-matrix consisting of all nonzero columns on k rows that do not haveˆ1 1˜i n rows 1 and 2. Let 0 denote the column of k 0's. Define F k (t) to be the concatenation of 0 with t + 1 copies of D12. We are able to show that forb(m, F k (t)) is Θ(m k−1 ). Linear algebra methods and indicator polynomials originated in this context in a paper of the authors and Füredi and Sali. We provide a novel application of these methods.The results are further evidence for the conjecture of Anstee and Sali on the asymptotics for fixed F of forb(m, F ).

Section: Corollary 12 Let K M Be Given and Let A Be A Simple M-rowmentioning

confidence: 60%

Linear algebra methods for forbidden configurations

Fleming

2011

Combinatorica

Self Cite

“…The major result of [4] yields that F B (s) and F 12 (s) are the maximal k-rowed matrices which have forb being O(m k−1 ). Here maximal means that adding a column not already present s times will result in a configurations with forb being Ω(m k ).…”

Section: Large Configurationsmentioning

confidence: 99%

“…We use this with u = m α . The following proof is in [4]. Theorem 3.9 Let k be given and B be a given k × (k + 1) simple matrix with one column of each column sum.…”

Section: Large Configurationsmentioning

confidence: 99%

Large forbidden configurations and design theory

Studia Scientiarum Mathematicarum Hungarica

Sali

2016

Let forb(m, F ) denote the maximum number of columns possible in a (0,1)-matrix A that has no repeated columns and has no submatrix which is a row and column permutation of F . We consider cases where the configuration F has a number of columns that grows with m. For a k × matrix G, define s · G to be the concatenation of s copies of G. In a number of cases we determine forb(m, m α · G) is Θ(m k+α ). Results of Keevash on the existence of designs provide constructions that provide asymptotic lower bounds. An induction idea of Anstee and Lu is useful in obtaining upper bounds.

“…By Lemma 4, if σ 1 (B) > 2m, we can find an arrangement of 9 1's as in (3). Let y 1 denote the central 1 with y 2 denoting the 1 below y 1 and y 3 denoting the 1 above.…”

Section: Submatrices Ofmentioning

confidence: 99%

“…We have noted (in [3]) that forb(m, {E 1 , E 2 , E 3 }) = 2. Balogh and Bollobás [7] have the much more general result that for a given k, there is a constant c k such that forb(m,…”

Section: Conjecturementioning

confidence: 99%

Forbidden Configurations and Product Constructions

Graphs and Combinatorics

Koch

Raggi

et al. 2013

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.