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On the density of matroids omitting a complete-graphic minor

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arxiv 2306.15061 v1 pith:F6SXSJS7 submitted 2023-06-26 math.CO

classification math.CO
keywords matroidminorratioaboveanswerbest-possibleboundbounded
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abstract

We show that, if $M$ is a simple rank-$n$ matroid with no $\ell$-point line minor and no minor isomorphic to the cycle matroid of a $t$-vertex complete graph, then the ratio $|M| / n$ is bounded above by a singly exponential function of $\ell$ and $t$. We also bound this ratio in the special case where $M$ is a frame matroid, obtaining an answer that is within a factor of two of best-possible.

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