The f-Factor Problem for Graphs and the Hereditary Property
  If P is a hereditary property then we show that, for the existence of a perfect f-factor, P is a sufficient condition for countable graphs and yields a sufficient condition for graphs of size aleph_1. Further we give two examples of a hereditary property which is even necessary for the existence of a perfect f-factor. We also discuss the aleph_2-case. 
Frank Niedermeyer and Saharon Shelah and Karsten Steffens