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According to Oxley's Matroid Theory, page 19 (second edition), a uniform matroid is a matroid with $n$ elements in set $E$, so that for some integer $m$ the set $I$ is defined: $I(U_{m,n})=\{X \subseteq E: |X|\leq m\}$.

But if the bases of some matroid are of the same cardinality, doesn't every matroid satisfy this characteristic? Or that if a matroid's bases are of cardinality $m$, of course every set of bigger cardinality is a cycle?

Maybe I'm just confused. Any assistance would be very appreciated! Also, English isn't my native language, so sorry if I didn't write things correctly, grammar-wise.

NoName
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Avi
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1 Answers1

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You're right that you can't have an independent set of cardinality greater than the rank. The definition of uniform matroid says that every set $X$ of cardinality less than or equal to the rank is independent. There are plenty of examples of matroids where this isn't the case. For instance, a matroid could have a loop $x$, such that ${x}$ is dependent. Then any set containing $x$ will also be dependent.

NoName
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