Extremal Functions of Forbidden 0-1 Matrices

A 0-1 matrix is a matrix with every entry equal to 0 or 1. A 0-1 matrix A avoids a pattern P given as a 0-1 matrix if no submatrix of A either equals P or can be transformed to P by changing some ones to zeroes.

Let ex(n, P) denote the maximum number of ones in a 0-1 matrix that avoids P. For any 0-1 matrix P with at least two ones, ex(n, P) >= n sice either the n x n 0-1 matrix with ones only in the first column or the n x n 0-1 matrix with ones only in the first row avoids P.

## Applications of ex(n, P)

Upper bounds on ex(n, P) for a family of 0-1 matrices P have been used to find upper bounds on the complexity of algorithms to find a minimal path in a rectilinear grid with obstacles.

Upper bounds on ex(n, P) for a different family of 0-1 matrices P have been used to bound the maximum number of unit distances in a convex n-gon.

For all permutation matrices P, ex(n, P) = O(n). This result was used to prove the Stanley-Wilf conjecture. The Stanley-Wilf conjecture states that if p is a permutation of 1, ..., k, then the number of permutations of 1, ..., n that avoid p has an upper bound of cn for some constant c.

Bounds on ex(n, P) have been used to find bounds on extremal functions of forbidden sequences. Extremal functions of forbidden sequences have been used to find bounds on the maximum complexity of lower envelopes of sets of polynomials of bounded degree, the maximum number of edges in k-quasiplanar graphs with no pair of edges intersecting in more than O(1) points, and the complexity of faces in arrangements of arcs. The function ex(n, P) can also be viewed as an extremal function for ordered bipartite graphs, where vertices correspond to rows and columns and edges correspond to ones.

## Useful Facts about ex(n, P)

This section includes some useful facts about ex(n, P) that have been discussed on the CrowdMath message board.

### Transformations that preserve ex(n, P)

For an integer n and a k x l matrix P, ex(n, P) = ex(n, P') if P' is the product of rotating, reflecting, and transposing P a given number of times. (Here, rotation means rotating by 90 degrees clockwise, reflecting means flipping about a horizontal or vertical axis, and transposing means taking the transpose.)

### Inequalities for ex(n, P) through containment

If P' contains P, then ex(n, P') >= ex(n, P).

Fact 0: Let P be a pattern, and let re(P) be the set of all matrices formed by deleting some or none of P's 1 entries. If P' is in re(P) and A avoids P', then A avoids P.

Corollary 1: Let Q be a pattern, and let ad(Q) be the set of all patterns created by adding some or no 1 entries to Q. If Q' is in ad(Q) and A avoids Q, then A avoids Q'.

Corollary 2: If Q is in re(P), ex(n, Q) <= ex(n, P) and if A is in ad(P), then ex(n, P) <= ex(n, A).

### Composition of Patterns

Suppose that P is a pattern with a 1 in its bottom left corner and Q is a pattern with a 1 in its top right corner. If R is obtained by joining P and Q so that the the bottom left one of P is the same as the top right one of Q, then max(ex(n, P), ex(n, Q)) <= ex(n, R) <= ex(n, P) + ex(n, Q).

## Bounds on ex(n, P)

The notation ex(n, P) = O(f(n)) means that there exists k such that ex(n, P) <= k f(n) for all n sufficiently large. It is an open problem to find every 0-1 matrix P for which ex(n, P) = O(n). P is called linear if ex(n, P) = O(n).

An equivalent problem is to identify all of the minimally non-linear 0-1 matrices. A 0-1 matrix P is called minimally non-linear if the following 2 conditions are true:

1. P is not linear
2. For every pattern P' contained in P for which P' is not P, P' is linear

It is known that there are infinitely many minimally non-linear 0-1 matrices, but only finitely many have been identified. It is an open problem to identify an infinite class of minimally non-linear 0-1 matrices.

## Bounds on exk(m, P)

A different extremal function exk(m, P) has been used to bound ex(n, P). The column extremal function exk(m, P) is the maximum number of columns in a 0-1 matrix with m rows that avoids P and has at least k ones in every column.

It is an open problem to find every patttern P for which exk(m, P) = O(m). Another open problem is to find every pattern P for which exk(m, P) = O(m/k).