Efficient Removal Lemmas for Matrices

It was recently proved in Alon et al. (2017) that any hereditary property of two-dimensional matrices (where the row and column order is not ignored) over a finite alphabet is testable with a constant number of queries, by establishing the following ordered matrix removal lemma: For any finite alphabet Γ, any hereditary property P of matrices over Γ, and any ϵ > 0, there exists f_P such that for any matrix M over Γ that is ϵ-far from satisfying P, most of the f_P × f_P submatrices of M do not satisfy P. Here being ϵ-far from P means that one needs to modify at least an ϵ-fraction of the entries of M to make it satisfy P. However, in the above general removal lemma, f_P grows very quickly as a function of ϵ^{− 1}, even when P is characterized by a single forbidden submatrix. In this work we establish much more efficient removal lemmas for several special cases of the above problem. In particular, we show the following, which can be seen as an efficient binary matrix analogue of the triangle removal lemma: For any fixed s × t binary matrix A and any ϵ > 0 there exists δ > 0 polynomial in ϵ, such that for any binary matrix M in which less than a δ-fraction of the s × t submatrices are equal to A, there exists a set of less than an ϵ-fraction of the entries of M that intersects every copy of A in M. We generalize the work of Alon et al. (2007) and make progress towards proving one of their conjectures. The proofs combine their efficient conditional regularity lemma for matrices with additional combinatorial and probabilistic ideas.