It is known that almost all matroids are not linear matroids (a.k.a. not representable matroids). This was shown by Nelson:
arXiv: Almost all matroids are non-representable
A transversal matroid is a certain type of linear matroid with various equivalent definitions. Transerval matroids are linear matroids, and they can always be represented over any infinite field. See the survey of Bonin for a nice introduction:
AN INTRODUCTION TO TRANSVERSAL MATROIDS
My question is how rare are transversal matroids amongst all linear matroids? I am most interested in real representable matroids, but interested in any results enumerating or estimating how many transversal matroids there are.
