Linear algebra methods for forbidden configurations

Abstract

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 ).

Large forbidden configurations and design theory

“…The proof for F 12 (m α ) again follows the proof in [5]. An open problem is to do the same analysis for the following configuration named the 'chestnut'.…”

“…Lemma 3.8 (Lemma 4.4 in [5]) Let A, k and u be given and assume that S denotes those k-sets of rows S of A for which A| S has at least two different k × 1 columns α, β with µ(α, A| S ) < u and µ(β, A| S ) < u. We can delete O(m k−1 ) columns from A to obtain A so that for each S ∈ S, there is some column not present in A| S .…”

Forbidden Configurations and Product Constructions

“…For any subset R(1) ⊂ [n 1 ] and R(2) ⊂ [n 2 ] we define A| (R(1),R(2)) as the submatrix of A formed of the entries contained in the rows R(1) and in the columns R (2). In this section we will be considering cases where both R(1) and R(2) consist of consecutive integers.…”

“…, Q 4 . These cases were computed using a C++ program (can be downloaded at [2]) that had many test runs checking correctness and was also independently checked by a program written in sage (public code that uses Python). In each case one may easily check that, if the case is satisfied, indeed our three matrices of (1) aren't present as configurations.…”