We deal with the problem of preserving various versions of completeness in (< κ)–support iterations of forcing notions, generalizing the case “S–complete proper is preserved by CS iterations for a stationary costationary S ⊆ ω1”. We give applications to Uniformization and the Whitehead problem. In particular, for a strongly inaccessible cardinal κ and a stationary set S ⊆ κ with fat complement ...