  In a 1977 paper, Steffens identified an elegant criterion for determining
when a countable graph has a perfect matching. In this paper, we will
investigate the proof-theoretic strength of this result and related theorems.
We show that a number of natural variants of these theorems are equivalent, or
closely related, to the ``big five'' subsystems of reverse mathematics.
  The results of this paper explore the relationship between graph theory and
logic by showing the way in which specific changes to a single graph-theoretic
principle impact the corresponding proof-theoretical strength. Taken together,
the results and questions of this paper suggest that the existence of matchings
in countable graphs provides a rich context for understanding reverse
mathematics more broadly.
