Abstract In [15] we defined and proved the consistency of the principle upper G upper M Superscript plus Baseline left parenthesis omega 3 comma omega 1 right parenthesis $\mathrm {GM}^+(\omega _3,\omega _1)$ G M + ( ω 3 , ω 1 ) which implies that many consequences of strong forcing axioms hold simultaneously at omega 2 $\omega _2$ ω 2 and omega 3 $\omega _3$ ω 3 . In this paper we formulate a strengthening of upper G upper M Superscript plus Baseline left parenthesis omega 3 comma omega 1 right parenthesis $\mathrm {GM}^+(\omega _3,\omega _1)$ G M + ( ω 3 , ω 1 ) that we call upper S upper G upper M Superscript plus Baseline left parenthesis omega 3 comma omega 1 right parenthesis $\mathrm {SGM}^+(\omega _3,\omega _1)$ S G M + ( ω 3 , ω 1 ) . We also prove, modulo the consistency of two supercompact cardinals, that upper S upper G upper M Superscript plus Baseline left parenthesis omega 3 comma omega 1 right parenthesis $\mathrm {SGM}^+(\omega _3,\omega _1)$

Abstract We prove that it is relatively consistent with ZF + CH that there exist two models of cardinality $\aleph _2 $ such that the second player has a winning strategy in the Ehrenfeucht–Fraïssé-game of length ω1 but there is no σ-closed back-and-forth set for the two models. If CH fails, no such pairs of models exist.

Abstract We show that it is relatively consistent with ZFC to have PCF structures of heightδ, for all ordinalsδ<ω3.

We describe efficient constructions of small probability spaces that approximate the independent distribution for general random variables. Previous work on efficient constructions concentrate on approximations of the independent distribution for the special case of uniform boolean-valued random variables. Our results yield efficient constructions of small sets with low discrepancy in high dimensional space and have applications to derandomizing randomized algorithms.

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In this paper we consider various properties of Jensen's □ principles and use them to construct several examples concerning the so-called Novák number of partially ordered sets. In §1 we give the relevant definitions and review some facts about □ principles. Apart from some simple observations most of the results in this section are known. In §2 we consider the Novák number of partially ordered sets and, using □ principles, give counterexamples to the productivity of this cardinal function. We also formulate a principle, show by forcing that it is consistent and use it to construct an ℵ2-Suslin tree T such that forcing with T × T collapses ℵ1. In §3 we briefly consider games played on partially ordered sets and relate them to the problems of the previous section. Using a version of □ we give an example of a proper partial order such that the game of length ω played on is undetermined. In §4 we raise the question of whether the Novák number of a homogenous partial order can be singular, and show that in some cases the answer is no. We assume familiarity with the basic techniques of forcing. In §1 some facts about large cardinals (e.g. weakly compact cardinals are -indescribable) and elementary properties of the constructible hierarchy are used. For this and all undefined terms we refer the reader to Jech [10].