Fractal classes of matroids
Combinatorics
2019-08-01 v1
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.
Keywords
Cite
@article{arxiv.1907.13343,
title = {Fractal classes of matroids},
author = {Dillon Mayhew and Mike Newman and Geoff Whittle},
journal= {arXiv preprint arXiv:1907.13343},
year = {2019}
}