Abstract
Matroids are combinatorial objects that abstract notions of "independence" in a multitude of areas in mathematics.
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. After giving an overview of random matroid theory, I will present recent progress in this direction: almost every matroid has at least one of two particular matroids, the rank-3 wheel and the rank-3 whirl, 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.
No prior knowledge about matroids is assumed.