Characterizing real-representable matroids with large average hyperplane-size

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, Grünbaum, Erdős 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.