On the one hand we deal with (< κ)-supported iterated forcing notions which are (Ê0, Ê1)-complete, have in mind problems on Whitehead groups, uniformizations and the general problem. We deal mainly with the caes of a successor of the singular cardinal. This continues [Sh 587]. On the other hand we deal with complimentary ZFC combinatorial results. I would like to thank Alice Leonhardt for the b...