Iteration of λ-complete forcing notions not collapsing λ

We look for a parallel to the notion of " proper forcing " among λ-complete forcing notions not collapsing λ +. We suggest such a definition and prove that it is preserved by suitable iterations. This work follows [Sh 587] and [Sh 667] (and see history there), but we do not rely on those papers. Our goal in this and the previous papers is to develop a theory parallel to " properness in CS iterations " for iterations with larger supports. In [Sh 587], [Sh 667] we have presented parallels to [Sh 64] and [Sh:98], whereas here we try to have parallels to [Sh 100], It seems too much to hope for a notion fully parallel to " proper " among λ-complete forcing notions as even for " λ +-c.c. λ-complete " there are problems. We should also remember about ZFC limitations for possible iteration theorems. For example, if in the definition of the forcing notion Q * in Section 3 we demand h p e δ ⊆ h δ , then the proof fails. This may seem a drawback, but one should look at [Sh:f, AP, p.985, 3.6(2) and p.990, 3.9]. By it, if S * = S λ + λ , and (A δ , h δ are as in 3.4 and) we ask a success on a club, then for some h δ : δ ∈ S λ + λ we fail. Now, if we allow only h δ : A δ −→ 2 and we ask for " success of the uniformization " on an end segment of A δ (for all such A δ : δ ∈ S λ + λ), then we also fail as we may code colourings with values in λ. In the first section we formulate our definitions (including properness over λ, see 1.3). We believe that our main Definition 1.3 is quite reasonable and applicable. One may also define a version of it where the diamond is " spread out ". The second section is devoted to the proof of the preservation theorem, and the next one gives three (relatively easy) examples of forcing notions fitting our scheme. We conclude the paper with the discussion of applications and variants.