Removal Lemmas for Matrices

The graph removal lemma is an important structural result in graph theory. It has many different variants and is closely related to property testing and other areas. Our main aim in this paper is to develop removal lemmas of the same spirit for two dimensional matrices. A matrix removal lemma is a statement of the following type: fix a finite family F of matrices over some alphabet Γ. Suppose that M is an n × n matrix over Γ, such that for any two positive integers s, t only o (n) of the s× t submatrices of M are equal to matrices from F . Then one can modify no more than o(n) entries in M to make it F -free (that is, after the modification no submatrix of M is equal to a matrix from F ). We prove matrix removal lemmas in several different scenarios. As a representative example, the following is one of our main results: fix an s × t binary matrix A. For any ǫ > 0 there exists δ > 0 such that for any n × n binary matrix M that contains less than δn copies of A as submatrices, there exists a set of ǫn entries of M that intersects every A-copy in M . Moreover, this removal lemma is efficient: δ is polynomial in ǫ. The major difficulty in our case is that the rows and the columns of a matrix are ordered. These are the first removal lemma type results for two dimensional graph-like objects with a predetermined order. Our results have direct consequences in property testing of matrices: they imply that for several types of families F and choices of the alphabet Γ, one can determine with good probability whether a given matrix M is F -free or ǫ-far from F -freeness (i.e. one needs to change at least an ǫ-fraction of its entries to make it F -free) by sampling only a constant number of entries in M . Our results generalize the work of Alon, Fischer and Newman [4] and make progress towards settling one of their open problems. The proofs combine a conditional regularity lemma for matrices proved in [4] with additional combinatorial and probabilistic ideas.