The joints problem for matroids

TL;DR

This paper establishes bounds on the maximum number of joints formed by lines in a simple matroid, showing it grows slower than quadratic but at least nearly quadratic in the number of lines.

Contribution

It proves new bounds on the number of joints in simple matroids, extending geometric joint problems to matroid theory.

Findings

Max number of joints is o(L^2) in simple matroids

Max number of joints is Omega(L^{2 - epsilon}) for any epsilon > 0

Bounds generalize geometric joint problem to matroids

Abstract

We prove that in a simple matroid, the maximal number of joints that can be formed by L lines is o(L^2) and Omega(L^{2 - epsilon}) for any epsilon > 0.