A Density Hales-Jewett Theorem for matroids

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