On the structure of matrices avoiding interval-minor patterns

Let Q be the 3×3 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×n having a row with Ω(n) alternating intervals of 0-entries and 1-entries.