The authors and Fischer recently proved 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 @math , any hereditary property @math of matrices over @math , and any @math , there exists @math such that for any matrix @math over @math that is @math -far from satisfying @math , most of the @math submatrices of @math do not satisfy @math . Here being @math -far from @math means that one needs to modify at least an @math -fraction of the entries of @math to make it satisfy @math . However, in the above general removal lemma, @math grows very fast as a function of @math , even when @math 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: For any fixed @math binary matrix @math and any @math there exists @math polynomial in @math , such that for any binary matrix @math in which less than a @math -fraction of the @math submatrices are equal to @math , there exists a set of less than an @math -fraction of the entries of @math that intersects every @math -copy in @math . We generalize the work of Alon, Fischer and Newman [SICOMP'07] 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.
We consider properties of edge-colored vertex-ordered graphs, i.e., graphs with a totally ordered vertex set and a finite set of possible edge colors. We show that any hereditary property of such graphs is strongly testable, i.e., testable with a constant number of queries. We also explain how the proof can be adapted to show that any hereditary property of @math -dimensional matrices over a finite alphabet (where row and column order is not ignored) is strongly testable. The first result generalizes the result of Alon and Shapira [FOCS'05, SICOMP'08], who showed that any hereditary graph property (without vertex order) is strongly testable. The second result answers and generalizes a conjecture of Alon, Fischer and Newman [SICOMP'07] concerning testing of matrix properties. The testability is proved by establishing a removal lemma for vertex-ordered graphs. It states that for any finite or infinite family @math of forbidden vertex-ordered graphs, and any @math , there exist @math and @math so that any vertex-ordered graph which is @math -far from being @math -free contains at least @math copies of some @math (with the correct vertex order) where @math . The proof bridges the gap between techniques related to the regularity lemma, used in the long chain of papers investigating graph testing, and string testing techniques. Along the way we develop a Ramsey-type lemma for @math -partite graphs with "undesirable" edges, stating that one can find a Ramsey-type structure in such a graph, in which the density of the undesirable edges is not much higher than the density of those edges in the graph.