The computational strength of matchings in countable graphs

Abstract

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.