Matroids of rank two I am interested in matroids of rank two and would like to understand how interesting/big this class of matroids is.
I know that the 2-uniform matroid on (k+2) elements is not representable over any field with at most k elements. (see Oxley, p203).
Does there exist any survey on matroids of rank two?
 A: Up to simplification (suppressing loops and parallel elements), every rank two matroid is just a rank two uniform matroid.  
Note that the vectors $(1, a_1), \dots, (1, a_n)$ represent the uniform matroid $U_{2,n}$, provided that $a_1, \dots, a_n$ are all distinct.  Thus, the rank two matroids are all representable over each infinite field.  
A: At first, neglect all loops (elements of rank 0). Now call two elements $a,b$ similar, if $\{a,b\}$ is not a base. It follows that if $\{a,c\}$ is a base, than $\{b,c\}$ is also a base (else the set $\{a,b,c\}$ would have inclusion-maximal independent subsets $\{b\}$ and $\{a,c\}$ of different size.) In other words, similarity is an equivalence relation. So, all elements may be partitioned onto groups so that any two elements of different groups form a base, and from the same group not. Of course such a matroid is representable over sufficiently large field. Namely, if we have $m$ groups, we should have $m$ different directions of the lines on the plane, that is, the field must contain at least $m-1$ elements (and this suffices).
