On the density of matroids omitting a complete-graphic minor

TL;DR

This paper establishes upper bounds on the density of simple rank-$n$ matroids avoiding certain minors, including complete-graphic minors, with specific bounds for frame matroids, advancing understanding of matroid density constraints.

Contribution

It provides new bounds on the density of matroids excluding complete-graphic minors, including a near-optimal bound for frame matroids.

Findings

Density ratio is bounded by a singly exponential function of $\

Bounds are tight within a factor of two for frame matroids.

Results extend previous minor exclusion density bounds to complete-graphic minors.

Abstract

We show that, if $M$ is a simple rank-$n$ matroid with no $\ell$-point line minor and no minor isomorphic to the cycle matroid of a $t$-vertex complete graph, then the ratio $|M| / n$ is bounded above by a singly exponential function of $\ell$ and $t$. We also bound this ratio in the special case where $M$ is a frame matroid, obtaining an answer that is within a factor of two of best-possible.