THOMAS JECH AND KAREL PRIKRY More generally, if £ is a nice cardinal function and $(00.,) is a strong limit cardinal, then 2 L D(u)2). Theorem 4.2.2. Assume that bt, carries a precipitous ideal, and let 0 be a nice cardinal function. If $(0)..) is a strong limit cardinal, then 2 L D((2 V ) . If $ is the enumeration of fixed points of the aleph function, then Theorem 4.2.2 gives an estimate for a case left open in [6], In section 5 we introduce the method of almost disjoint functions. Section 6 contains elementary proofs of the theorems from sections 3 and 4. In some cases we get slightly harper results than those obtained with generic ultrapowers. For instance, the elementary proof of Theorem 3.2.1 gives the following result on almost disjoint sets. Theorem 6.1.1. Assume that 2 2 and that 2 K . Then W l a) There exists an almost disjoint family of size 2 of uncountable subsets of OJ1 . N i b) If I is a a-complete ideal over LO , then for every X 2 there exists an almost disjoint family of size A of sets not in I. Con- *i sequently, sat(I) 2 Part a) of Theorem 6.1.1 was proved several years ago by Baumgartner, see [1]. In section 7 we obtain estimates on 2 and 2 under the assumption that there exists a nonregular ultrafilter over co1 :

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