2 THOMAS JECH AND KAREL PRIKRY

The method of almost disjoint functions is a refinement of the method of

almost disjoint transversals studied by combinatorial set theorists (e.g. [4]).

Given an ideal I over an uncountable set, we investigate the number of par-

tial ordinal functions (modulo equality almost everywhere) with prescribed

range of values. We evaluate sizes of families of partial functions. Among

others we obtain results on sizes of almost disjoint families of sets which

are not in the ideal. These results generalize some earlier results on almost

disjoint families of uncountable sets, and give a new insight into the well

known open problem whether the closed unbounded filter on c o can be ^ -sat-

urated.

Following Silver's work [14] on the singular cardinals problem, Baumgart-

ner, Jensen and Prikry found an elementary proof of Silver's theorem (cf. [2]),

using the method of almost disjoint transversals. Subsequently, Galvin and

Hajnal [6] obtained further results on the singular cardinals problem, using a

generalization of the elementary proof of Silver's theorem. We further refine

the method of Galvin and Hajnal, to give elementary proofs of our results on

the generalized continuum hypothesis and the singular cardinals problem.

As a byproduct of our method we obtain some consequences of existence of

nonregular ultrafilters for cardinal exponentiation.

Many of our results are proved under the assumption that there exists a

precipitous ideal. We have arrived at the notion of precipitous ideal as we

tried to extend the result of Galvin and Hajnal for singular cardinals of the

kind ^ = a. In view of equiconsistency of precipitous ideals and measurable

a

cardinals (cf. [7]) it appears that a precipitous ideal over GO is the exact

counterpart of measurability, suitable for c o .

The introductory section 1 contains the relevant definitions. In section

2 we introduce the method of generic ultrapowers and prove some basic lemmas.

Section 3 contains a number of results in which saturation of ideals over

K

a regular uncountable cardinal K is used to evaluate 2 . Among the theorems