A counting lemma for binary matroids and applications to extremal problems

Abstract

In graph theory, the Szemerédi regularity lemma gives a decomposition of the indicator function for any graph G into a structured component, a uniform part, and a small error. This result, in conjunction with a counting lemma that guarantees many copies of a subgraph H provided a copy of H appears in the structured component, is used in many applications to extremal problems. An analogous decomposition theorem exists for functions over Fpn. Specializing to p=2, we obtain a statement about the indicator functions of simple binary matroids. In this paper we extend previous results to prove a corresponding counting lemma for binary matroids. We then apply this counting lemma to give simple proofs of some known extremal results, analogous to the proofs of their graph-theoretic counterparts, and discuss how to use similar methods to attack a problem concerning the critical numbers of dense binary matroids avoiding a fixed submatroid.