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Can anyone tell me an example of a matroid of rank 5 or higher which is not a product of two lower rank matroids and is not the independence matroid of a finite set of vectors in a vector space over any field?

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Any irreducible matroid of rank 5 or higher that contains the Vamos matroid (en.wikipedia.org/wiki/Vámos_matroid) as a minor will do. –  Richard Stanley Dec 9 at 1:25

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