Strongly Maximal Matchings in Infinite Graphs - Aharoni, Ron; Berger, Eli; Georgakopoulos, Agelos; Sprüssel, Philipp (The Electronic Journal of Combinatorics, 2008)
Given an assignment of weights w to the edges of an infinite graph G, a matching M in G is called strongly w-maximal if for any matching N there holds {w(e) e N M} {w(e) e M N}. We prove that if w assumes only finitely many values all of which are rational then G has a strongly w-maximal matching. This result is best possible in the sense that if we allow irrational values or infinitely many values then there need not be a strongly w-maximal matching.