Matroid minors

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This proposal is related to a fundamental problem in combinatorial geometry posed by Whitney in his seminal paper on matroid theorey in 1935. We are interested in embedding points from a finite set E into a space (more specifically, a vector space over a field F) such that each subset A of E spans a subspace of some prescribed dimension rk(A). Such an embedding is called a representation of (E, rk) over F. Whitney asked which such pairs (E, rk) admit embeddings over F. He gave necessary conditions for the existence of such embeddings, and coined the term "matroid" for the pairs (E, rk) that satisfy these conditions. Matroid theory has since grown into a highly active research area with significant applications in coding theory, information theory, combinatorial optimization, electrical engineering, and theoretical computer science.