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The Princeton Companion to Mathematics by Imre Leader, June Barrow-Green, Timothy Gowers

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III.54 Matroids

Dominic Welsh

The original aim of Hassler Whitney when he introduced the concept of a matroid in 1935 was to produce an abstract notion that would capture the main ingredients of the structure of a set of vectors in a VECTOR SPACE [I.3 §2.3], while avoiding any explicit mention of linear independence.

To do this he singled out two fundamental properties and postulated that any family of subsets that possessed these properties was the collection of “independent sets” of a “matroid.” The first of these properties was an obvious one: any subset of a linearly independent set is also linearly independent. The second property was more subtle: if A and B are two linearly independent sets and B contains more elements than A, then there ...

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