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arXiv · 1409.0777

Matroids denser than a clique

Jim Geelen·Peter Nelson
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Abstract

The growth-rate function for a minor-closed class $\mathcal{M}$ of matroids is the function $h$ where, for each non-negative integer $r$, $h(r)$ is the maximum number of elements of a simple matroid in $\mathcal{M}$ with rank at most $r$. The Growth-Rate Theorem of Geelen, Kabell, Kung, and Whittle shows, essentially, that the growth-rate function is always either linear, quadratic, exponential, or infinite. Morover, if the growth-rate function is quadratic, then $h(r)\ge \binom{r+1}{2}$, with the lower bound coming from the fact that such classes necessarily contain all graphic matroids. We characterise the classes that satisfy $h(r) = \binom{r+1}{2}$ for all sufficiently large $r$.

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Jim Geelen, Peter Nelson. 2014-09-02. Matroids denser than a clique. https://arxiv.org/abs/1409.0777

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