A compactness theorem for singular cardinals, free algebras, Whitehead problem and tranversals

We prove, in an axiomatic way, a compactness theorem for singular cardinals. We apply it to prove that, for singular λ, every λ-free algebra is free; and similar compactness results for transversals and colouring numbers. For the general result on free algebras, we develop some filters on S _k(A). As an application we conclude that V=L implies that every Whitehead group is free.