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Abstract. 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.

Abstract. We use the method of norms on possibilities to answer a question of Kunen and construct a ccc σ-ideal on 2 ω with various closure properties and distinct from the ideal of null sets, the ideal of meager sets and their intersection. Introduction.In the present paper we use the method of norms on possibilities to answer a question of Kunen (see 0.1 below) and construct a ccc σ-ideal on 2 ω with various closure properties and distinct from the ideal of null sets, the ideal of meager sets and their intersection. The method we use is, in a sense, a generalization of the one studied systematically in [13] (the case of creating pairs). However, as the main desired property of the forcing notion we construct is satisfying the ccc, we do not use the technology of that paper (where the forcing notions were naturally proper not-ccc) and our presentation does not require familiarity with the previous part.

In the present paper we study the ideal of all subsets of 3?°> for which the second player has a winning strategy in the unsymmetric game. We describe its cardinal coefficients and the notions of forcing determined by it.

We continue developing the general theory of forcing notions built with the use of norms on possibilities, this time concentrating on ccc forcing notions and classifying them.