Abstract
We study the structure of 01-matrices avoiding a pattern P as an interval minor. We focus on critical P-avoiders, i.e., on the P-avoiding matrices in which changing a 0-entry to a 1-entry always creates a copy of P as an interval minor.
Let Q be the 3 x 3 permutation matrix corresponding to the permutation 231. As our main result, we show that for every pattern P that has no rotated copy of Q as interval minor, there is a constant cp such that any row and any column in any critical P-avoiding matrix can be partitioned into at most cp intervals, each consisting entirely of 0-entries or entirely of 1-entries.
In contrast, for any pattern P that contains a rotated copy of Q, we construct critical P-avoiding matrices of arbitrary size n x n having a row with Omega(n) alternating intervals of 0-entries and 1-entries.