The Taub Faculty of Computer Science Events and Talks

The triangle removal lemma, proved by Ruzsa and Szemerédi in 1976, states that if a graph contains a small number of triangles then it can be made triangle-free by a small number of edge deletions. The triangle removal lemma found applications in several areas of computer science and mathematics.

In a series of works, culminating in a result of Alon and Shapira from 2005, it was shown that in fact, any hereditary graph property P satisfies a removal lemma of this type: If most subgraphs of G of a certain fixed size satisfy P, then we can make G satisfy P using a small number of edge insertions/deletions.

However, these results only applied for unordered graphs, as the proof methods relied heavily on the fact that in such graphs there is no order on the vertices. For ordered structures, such as vertex-ordered graphs and matrices (whose rows and columns are ordered), no removal lemmas have been known.

In this talk we describe how to settle this issue, establishing an "order preserving" removal lemma for all hereditary properties of ordered graphs and matrices (and more generally, for all 2D structures that are somewhat well-behaved).

The result has direct implications in property testing, showing that any hereditary property of these ordered structures is constant-query testable with one-sided error.

Based on a joint work with Noga Alon and Eldar Fischer, that recently appeared in FOCS 2017.