Matrices with forbidden subconfigurations

Erdős-Szekeres Theorem for Point Sets with Forbidden Subconfigurations

According to the Erdős-Szekeres theorem, every set of n points in the plane contains roughly logn in convex position. We investigate how this bound changes if our point set does not contain a subset that belongs to a fixed order type.

Extremal functions of forbidden multidimensional matrices

Pattern avoidance is a central topic in graph theory and combinatorics. Pattern avoidance in matrices has applications in computer science and engineering, such as robot motion planning and VLSI circuit design. A d-dimensional zero-one matrix A avoids another d-dimensional zero-one matrix P if no submatrix of A can be transformed to P by changing some ones to zeros. A fundamental problem is to ...

Posets of Matrices and Permutations with Forbidden Subsequences

The enumeration of permutations with specific forbidden subsequences has applications in areas ranging from algebraic geometry to the study of sorting algorithms. We consider a ranked poset of permutation matrices whose global structure incorporates the solution to the equivalent problem of enumerating permutations which contain a required subsequence. We describe this structure completely for ...

Forbidden submatrices in 0-1 matrices

On the Column Extremal Functions of Forbidden 0-1 Matrices

A 0-1 matrix is a matrix in which every element is either 0 or 1. The weight extremal function ex(n, P ) counts the maximum number of 1’s in an n × n matrix which avoids a pattern matrix P . The column extremal function exk(m,P ) counts the maximum number of columns that a matrix with m rows and k 1’s per column can contain such that the matrix avoids P . Set weight and column extremal function...