Maximum size binary matroids with no AG(3,2)-minor are graphic

TL;DR

This paper characterizes the maximum size of simple binary matroids without an AG(3,2)-minor, showing that for rank at least 5, the maximum is inom{r+1}{2}nd identifying the unique or special matroids that achieve this bound.

Contribution

It establishes the maximum size of such matroids and characterizes all extremal cases, including the unique graphic matroid for ranks  and smaller examples.

Findings

Maximum size of binary matroids without AG(3,2)-minor is inom{r+1}{2}or rank  and above.

The graphic matroid M(K_{r+1}) uniquely attains this maximum for nd higher.

Characterization of maximum size non-regular binary matroids for each rank.

Abstract

We prove that the maximum size of a simple binary matroid of rank $r \geq 5$ with no AG(3,2)-minor is $\binom{r+1}{2}$ and characterise those matroids achieving this bound. When $r \geq 6$, the graphic matroid $M(K_{r+1})$ is the unique matroid meeting the bound, but there are a handful of smaller examples. In addition, we determine the size function for non-regular simple binary matroids with no AG(3,2)-minor and characterise the matroids of maximum size for each rank.