A Density Hales-Jewett Theorem for matroids

Abstract

We show that, if $\hat{I}\pm >0$ 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 $|M|\ge \hat{I}\pm q^{r(M)}$, 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.