Absolutely rigid systems and absolutely indecomposable groups
  We give a new proof that there are arbitrarily large indecomposable abelian groups; moreover, the groups constructed are absolutely indecomposable, that is, they remain indecomposable in any generic extension. However, any absolutely rigid family of groups has cardinality less than the partition cardinal kappa(omega). 
Paul C. Eklof and Saharon Shelah