A transversal matroid whose dual is not transversal In Oxley's Matroid Theory, Problem 14.8.5, it states that it is (or at least was in 1992) an open problem to determine when the dual matroid of a transversal matroid is also transversal. I had assumed that this did not hold for all transversal matroids, however I've had trouble finding an example of a transversal matroid whose dual is not transversal. Are the known examples of such matroids, and could someone point me towards one?
 A: Consider the rank-$4$ transversal matroid, $M$, shown in the diagram below. The figure on the left shows a presentation of $M$ via a bipartite graph. The ground set of $M$ is $\{a,b,c,d,e,f\}$. The figure on the right shows a geometric representation of $M$ in rank-$4$ space.

The dual matroid, $M^{*}$, has rank two. A geometric representation is shown below.
The easiest way to see that $M^{*}$ is non-transversal is to think of the geometric interpretation of transversal matroids: such a matroid is constructed by placing the points in the ground set in general position on the facets of a simplex. This means that if two points are placed at the same location, then they must be placed on the vertices of the simplex (the $0$-dimensional facets). But in rank $2$, there are only two vertices, which means that it is impossible to have three distinct pairs of points placed in identical locations.
If you don't like this geometrical argument, you could use the Mason-Ingleton characterisation of transversal matroids (Theorem 3.1 in Joe Bonin's Introduction to Transversal Matroids). We let $\mathcal{F}$ be $\{\{a,b\},\{c,d\},\{e,f\}\}$, and then when we apply the inequality we obtain $0$ on the left-hand side, and $-0+1+1+1-2-2-2+2=-1$ on the right, which certifies that $M^{*}$ is not transversal.
