  We prove that any non-amenable Cayley graph admits a factor of IID perfect
matching. We also show that any connected d-regular vertex tran- sitive graph
admits a perfect matching. The two results together imply that every Cayley
graph admits an invariant random perfect matching. A key step in the proof is a
result on graphings that also applies to finite graphs. The finite version says
that for any partial matching of a finite regular graph that is a good
expander, one can always find an augmenting path whose length is
poly-logarithmic in one over the ratio of unmatched vertices.
