8

THOMAS JECH AND KAREL PRIKRY

If I is prime then sat(I) = 2. If sat(I) is a finite number then K

is the union of finitely many atoms of I (a set X £ I is an atom if X/I

is an atom of the Boolean algebra B),and I is essentially a combination of

finitely many prime ideals.

If sat(I) is infinite, then by [5], sat(I) is a regular uncountable

cardinal. Clearly, every ideal I over K is (2 ) -saturated, and so if I

is atomless then sat(I) is a regular cardinal satisfying

(1.5) ^ sat(I) 5

(2K)+.

If K is real-valued measurable, then the ideal (1.2) is ^1-saturated.

If K carries a K-saturated ideal then K is weakly inaccessible (in fact

weakly Mahlo), see [15]. If K carries a K -saturated ideal I, then

carries a normal K -saturated ideal J such that sat(J) 5 sat(I), cf. [15].

Existence of a K -saturated ideal over K does not necessarily entail that

K is a limit cardinal: Kunen constructed a model in which tf carries an

^-saturated ideal.

The ideal of thin sets is not K-saturated: by a theorem of Solovay in

[15], every stationary subset of K is the union of K pairwise disjoint

stationary subsets. In section 4 we address ourselves to the question whether

the ideal of thin sets can be K -saturated (see the remark following Theorem

3.2.1.)

1.3. Two sets X £ K and Y £ K are almost disjoint if |X D Y| K. Fam-

ilies of almost disjoint sets have been investigated by Sierpinski, Tarski,

and more recently by Baumgartner [1]. It is clear that the question of size

of almost disjoint families of sets X £ K of size K is equivalent to the

problem of evaluating sat(I) for the ideal I = {X £ K : |X| K}.

It is easy to construct an almost disjoint family of K subsets of K

+ a

size K and so sat(I) K . If 2 K for all a K, then there exists