A Counting Lemma for Binary Matroids and Applications to Extremal Problems

TL;DR

This paper extends a decomposition and counting lemma from graph theory to binary matroids, enabling simpler proofs of extremal results and advancing understanding of dense binary matroids avoiding specific submatroids.

Contribution

It introduces a counting lemma for binary matroids, analogous to the graph regularity lemma, and applies it to derive extremal results and address problems on critical numbers.

Findings

Established a counting lemma for binary matroids.

Provided simplified proofs of known extremal results.

Discussed methods for analyzing dense binary matroids avoiding fixed submatroids.

Abstract

In graph theory, the Szemer\'edi 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 $\mathbb{F}_p^n$. 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.