On nonlinear forbidden 0-1 matrices: a refutation of a Füredi-Hajnal conjecture

  • Authors:

  • Seth Pettie

  • Affiliations:

  • University of Michigan

  • Venue:
  • SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
  • Year:
  • 2010

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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. Our primary contribution is a refutation of a conjecture of Füredi and Hajnal: that if P corresponds to an acyclic graph then the maximum number of 1s in an n x n matrix avoiding P is O(n log n). In addition, we give a simpler proof that there are infinitely many minimal nonlinear patterns and give tight bounds on the extremal functions for several small forbidden patterns.