  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 minor that is isomorphic to the cycle matroid of the
complete graph on four vertices. For each k that is at least 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)-base-orderable; the union of these sets, over
all k, is an infinite set of base-orderable excluded minors for strong
base-orderability.

Combinatorics