When does almost free imply free? (For groups, transversals, etc.)

Abstract:

We show that the construction of an almost free nonfree Abelian group can be pushed from a regular cardinal $\kappa$ to ${\aleph _{\kappa + 1}}$. Hence there are unboundedly many almost free nonfree Abelian groups below the first cardinal fixed point. We give a sufficient condition for “ $\kappa$ free implies free”, and then we show, assuming the consistency of infinitely many supercompacts, that one can have a model of ZFC+G.C.H. in which ${\aleph _{{\omega ^2} + 1}}$ free implies ${\aleph _{{\omega ^2} + 2}}$ free. Similar construction yields a model in which ${\aleph _\kappa }$ free implies free for $\kappa$ the first cardinal fixed point (namely, the first cardinal $\alpha$ satisfying $\alpha = {\aleph _\alpha }$). The absolute results about the existence of almost free nonfree groups require only minimal knowledge of set theory. Also, no knowledge of metamathematics is required for reading the section on the combinatorial principle used to show that almost free implies free. The consistency of the combinatorial principle requires acquaintance with forcing techniques.