Uniform matroid ($U_{r,n}$)

For $0\le r\le n$, the uniform matroid $U_{r,n}$ on an $n$-element ground set has as its matroid independent sets exactly the subsets of cardinality at most $r$. Heredity is immediate, and augmentation works because any smaller such subset can acquire an element of a larger one without exceeding $r$. Its rank of a matroid on $A$ is $\min \{|A|,r\}$.