A combinatorial principle equivalent to the existence of non-free   Whitehead groups
  As a consequence of identifying the principle described in the title, we prove that for any uncountable cardinal lambda, if there is a lambda-free Whitehead group of cardinality lambda which is not free, then there are many ``nice'' Whitehead groups of cardinality lambda which are not free. 
Paul C. Eklof, Saharon Shelah