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arXiv · 1206.6270

On the number of matroids

N. Bansal·R. A. Pendavingh·J. G. van der Pol
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Abstract

We consider the problem of determining $m_n$, the number of matroids on $n$ elements. The best known lower bound on $m_n$ is due to Knuth (1974) who showed that $\log \log m_n$ is at least $n-3/2\log n-1$. On the other hand, Piff (1973) showed that $\log\log m_n\leq n-\log n+\log\log n +O(1)$, and it has been conjectured since that the right answer is perhaps closer to Knuth's bound. We show that this is indeed the case, and prove an upper bound on $\log\log m_n$ that is within an additive $1+o(1)$ term of Knuth's lower bound. Our proof is based on using some structural properties of non-bases in a matroid together with some properties of independent sets in the Johnson graph to give a compressed representation of matroids.

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N. Bansal, R. A. Pendavingh, J. G. van der Pol. 2013-09-24. On the number of matroids. https://arxiv.org/abs/1206.6270

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