A Density Hales-Jewett Theorem for matroids

TL;DR

This paper proves a density version of the Hales-Jewett theorem for matroids, showing that large dense matroids avoiding certain minors contain affine geometry restrictions, extending combinatorial density results to matroid theory.

Contribution

It establishes a density Hales-Jewett type theorem for matroids, linking matroid minors, density, and affine geometries over finite fields.

Findings

Large dense matroids without specific minors contain affine geometry restrictions.

The result extends combinatorial density theorems to matroid structures.

Provides a new perspective on matroid minors and density conditions.

Abstract

We show that, if $\alpha > 0$ is a real number, $n \ge 2$ and $\ell \ge 2$ are integers, and $q$ is a prime power, then every simple matroid $M$ of sufficiently large rank, with no $U_{2,\ell}$-minor, no rank-$n$ projective geometry minor over a larger field than $\GF(q)$, and satisfying $|M| \ge \alpha q^{r(M)}$, has a rank-$n$ affine geometry restriction over $\GF(q)$. This result can be viewed as an analogue of the Multidimensional Density Hales-Jewett Theorem for matroids.