Degrees of nonlinearity in forbidden 0–1 matrix problems

Abstract

A 0–1 matrix A is said to avoid a forbidden 0–1 matrix (or pattern) P if no submatrix of A matches P, where a 0 in P matches either 0 or 1 in A. The theory of forbidden matrices subsumes many extremal problems in combinatorics and graph theory such as bounding the length of Davenport–Schinzel sequences and their generalizations, Stanley and Wilf’s permutation avoidance problem, and Turán-type subgraph avoidance problems. In addition, forbidden matrix theory has proved to be a powerful tool in discrete geometry and the analysis of both geometric and non-geometric algorithms.Clearly a 0–1 matrix can be interpreted as the incidence matrix of a bipartite graph in which vertices on each side of the partition are ordered. Füredi and Hajnal conjectured that if P corresponds to an acyclic graph then the maximum weight (number of 1s) in an n×n matrix avoiding P is O(nlogn). In the first part of the article we refute of this conjecture. We exhibit n×n matrices with weight Θ(nlognloglogn) that avoid a relatively small acyclic matrix. The matrices are constructed via two complementary composition operations for 0–1 matrices. In the second part of the article we simplify one aspect of Keszegh and Geneson’s proof that there are infinitely many minimal nonlinear forbidden 0–1 matrices. In the last part of the article we investigate the relationship between 0–1 matrices and generalized Davenport–Schinzel sequences. We prove that all forbidden subsequences formed by concatenating two permutations have a linear extremal function.