Characterizing real-representable matroids with large average hyperplane-size

TL;DR

This paper characterizes real-representable matroids with large average hyperplane size, extending classical theorems, and introduces a high-dimensional generalization of a geometric problem related to colored point sets.

Contribution

It extends known results on hyperplane sizes in real-representable matroids to complex and orientable cases, and formulates a new high-dimensional problem generalizing a classic geometric question.

Findings

Either the average hyperplane size is bounded by a constant depending on rank.

If not, the matroid's ground set can be partitioned with specific properties.

A high-dimensional problem related to colored points and monochromatic lines is formulated and analyzed.

Abstract

Generalizing a theorem of the first two authors and Geelen for planes, we show that, for a real-representable matroid $M$, either the average hyperplane-size in $M$ is at most a constant depending only on its rank, or each hyperplane of $M$ contains one of a set of at most $r(M)-2$ lines. Additionally, in the latter case, the ground set of $M$ has a partition $(E_{1}, E_{2})$, where $E_{1}$ can be covered by few flats of relatively low rank and $|E_{2}|$ is bounded. These results extend to complex-representable and orientable matroids. Finally, we formulate a high-dimensional generalization of a classic problem of Motzkin, Gr\"unbaum, Erd\H{o}s and Purdy on sets of red and blue points in the plane with no monochromatic blue line. We show that the solution to this problem gives a tight upper bound on $|E_{2}|$. We also discuss this high-dimensional problem in its own right, and prove some initial results.