Almost every matroid has a rank-3 wheel or rank-3 whirl as minor

Many conjectures – but few results – exist for the statistical properties of large "random" matroids. For example, the question of which matroids appear as a minor of almost every matroid has been settled for only a few matroids. I will present recent progress in this direction: almost every matroid has at least one of two particular matroids, the rank-3 wheel M(K_4) or the rank-3 whirl W^3, as a minor.

At the heart of the argument lies a counting version of the Ruzsa–Szemerédi (6,3)-theorem on 3-uniform hypergraphs, which is then generalised in several ways to obtain the main result.

This talk focuses on the hypergraph side of things; in particular, no prior knowledge about matroids is assumed.

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