Advances in Applied Mathematics ( IF 0.952 ) Pub Date : 2020-01-10 , DOI: 10.1016/j.aam.2019.101995
Dillon Mayhew; Mike Newman; Geoff Whittle

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 $\mathbb{K}$ is an infinite field, the class of $\mathbb{K}$-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 $k\ge 5$) 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.

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