Finite strict gammoids, introduced in the early 1970s, are matroids defined via finite digraphs equipped with some set of sinks: a set of vertices is independent if it admits a linkage to these sinks. An independent set is maximal precisely if it adm
its a linkage onto the sinks. In the infinite setting, this characterization of the maximal independent sets need not hold. We identify a type of substructure as the unique obstruction to the characterization. We then show that the sets linkable onto the sinks form the bases of a (possibly non-finitary) matroid precisely when the substructure does not occur.
We consider numbers and sizes of independent sets in graphs with minimum degree at least $d$, when the number $n$ of vertices is large. In particular we investigate which of these graphs yield the maximum numbers of independent sets of different size
s, and which yield the largest random independent sets. We establish a strengthened form of a conjecture of Galvin concerning the first of these topics.
