Matroids with no $U_{2,n}$-minor and many hyperplanes

TL;DR

This paper constructs specific matroids of any rank and prime power size that lack certain minors but have more hyperplanes than the corresponding projective geometries, revealing new extremal properties.

Contribution

It introduces a method to build matroids with no $U_{2,q+2}$-minor that surpass the hyperplane count of classical projective geometries.

Findings

Existence of such matroids for all ranks $r \\ge 3$ and prime powers $q > 10$.

Construction of matroids exceeding the hyperplane count of projective geometries.

Demonstration of extremal hyperplane properties in minor-closed classes.

Abstract

We construct, for every $r \ge 3$ and every prime power $q > 10$, a rank-$r$ matroid with no $U_{2,q+2}$-minor, having more hyperplanes than the rank-$r$ projective geometry over $\mathrm{GF}(q)$.