# Fractal classes of matroids

## TL;DR

This paper explores fractal properties of minor-closed matroid classes, proving certain classes are strongly fractal and analyzing the distribution of matroids and excluded minors within these classes.

## Contribution

It introduces the concept of fractal classes of matroids, proves that sparse paving matroids with limited circuit-hyperplanes are strongly fractal, and analyzes excluded minors in spike classes.

## Key findings

- Sparse paving matroids with up to three circuit-hyperplanes are strongly fractal.
- The class of spikes with more than four circuit-hyperplanes satisfies a weaker domination condition.
- Finitely many excluded minors exist with odd-sized ground sets.

## Abstract

A minor-closed class of matroids is (strongly) fractal if the number of n-element matroids in the class is dominated by the number of n-element excluded minors. We conjecture that when K is an infinite field, the class of K-representable matroids is strongly fractal. We prove that the class of sparse paving matroids with at most k circuit-hyperplanes is a strongly fractal class when k is at least three. The minor-closure of the class of spikes with at most k circuit-hyperplanes (with k>4) satisfies a strictly weaker condition: the number of 2t-element matroids in the class is dominated by the number of 2t-element excluded minors. However, there are only finitely many excluded minors with ground sets of odd size.