Degrees of nonlinearity in forbidden 0-1 matrix problems

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(n log n). Our first result is a refutation of this conjecture. We exhibiting n × n matrices with weight Θ(n log n log log n) that avoid a relatively small acyclic matrix. The matrices are constructed via two complementary composition operations for 0-1 matrices. Our second result is a simplified proof that there is an infinite antichain (with respect to containment) of minimally nonlinear forbidden matrices, which, as a byproduct, yields tight bounds on several previously unclassified forbidden matrices. In the final part of the article we present several results on the relationship between forbidden matrix theory and the theory of generalized Davenport-Schinzel sequences, which yields new sets of forbidden matrices with quasilinear extremal functions and new forbidden subsequences with a linear extremal function. ∗This work is supported by NSF CAREER grant no. CCF-0746673.