Pure pairs. VII. Homogeneous submatrices in 0/1-matrices with a forbidden submatrix

Forbidden submatrices in 0-1 matrices

On linear forbidden submatrices

In this paper we study the extremal problem of finding how many 1 entries an n by n 0-1 matrix can have if it does not contain certain forbidden patterns as submatrices. We call the number of 1 entries of a 0-1 matrix its weight. The extremal function of a pattern is the maximum weight of an n by n 0-1 matrix that does not contain this pattern as a submatrix. We call a pattern (a 0-1 matrix) li...

Avoiding Forbidden Submatrices by Row Deletions

We initiate a systematic study of the Row Deletion(B) problem on matrices: For a fixed “forbidden submatrix” B, the question is, given an input matrix A (both A and B have entries chosen from a finite-size alphabet), to remove a minimum number of rows such that A has no submatrix which is equivalent to a row or column permutation of B. An application of this question can be found, e.g., in the ...

Principal Submatrices VII : Further Results Concerning Matrices With Equal Principal Minors *

This paper characte rizes real symmetri c matri ces A s uch that all t X t princ ipal minors a re equal and a ll t X t nonprinc ipal minors a re of fixed sign, for two consecutive values of t less than rank A. It a lso cha rac te rizes matrices A (over an arbitrary fi eld ) in which a ll t X t principal minors are equal and all nonprin cipal t X t minors a re e qua l, for one fixed value of t l...

Forbidden Submatrices: Some New Bounds and Constructions

We explore an extremal hypergraph problem for which both the vertices and edges are ordered. Given a hypergraph F (not necessarily simple), we consider how many edges a simple hypergraph (no repeated edges) on m vertices can have while forbidding F as a subhypergraph where both hypergraphs have fixed vertex and edge orderings. A hypergraph of n edges on m vertices can be encoded as an m × n (0,...