An infinite family of excluded minors for strong base-orderability

We discuss a conjecture of Ingleton on excluded minors for base-orderability, and, extending a result he stated, we prove that infinitely many of the matroids that he identified are excluded minors for base-orderability, as well as for the class of gammoids. We prove that a paving matroid is base-orderable if and only if it has no M(K4)M(K4)-minor. For each k≥2k≥2, we define the property of k-base-orderability, which lies strictly between base-orderability and strong base-orderability, and we show that k-base-orderable matroids form what Ingleton called a complete class. By generalizing an example of Ingleton, we construct a set of matroids, each of which is an excluded minor for k  -base-orderability, but is (k−1)(k−1)-base-orderable; the union of these sets, over all k, is an infinite set of base-orderable excluded minors for strong base-orderability.