Reference for characterizations of the uniform matroid Let $m,n \in \mathbb N$ be natural numbers. The matroid $U_{m,n}$ on ground set $\{1,...,n\}$ with set of bases $\mathcal B(U_{m,n}) = \{X \subseteq \{1,...,n\}: |X| = m\}$ is called uniform matroid. 
There are different characterizations of this type of matroid, e.g., 


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*A matroid is uniform if and only if every circuit meets every cocircuit.

*A matroid is uniform if and only if every two of its elements are clones.

*A matroid is uniform if and only if it has no circuits of cardinality less than $r+1$, where $r$ is the rank of the matroid.


Question: Is there an article about uniform matroids which features more characterizations (including proofs)?
 A: Without doing an extensive search, I'm inclined to say no, because uniform matroids aren't really a subject of research. I've heard it said that uniform matroids aren't really matroids but pairs of integers (or more precisely, an integer and a set of finite cardinality not less than the integer). While there no doubt exist many succinct characterizations of uniform matroids, I don't know why someone would compile them in an article.
Uniformity certainly isn't the only property that has numerous characterizations in matroid theory. For instance, you can characterize matroid connectivity by saying that for every pair of elements there's a circuit using them, or there's not a partition of the ground set such that every basis is a basis of one side with a basis of the other side, or for every pair of elements there's a cocircuit using them... I would expect an article to say the ones they want to use, rather than listing all of them.
I suppose you might find notes that give novel characterizations of uniform matroids with proofs, such as this one.
