IDEALS OVER UNCOUNTABLE SETS 9

an almost disjoint family of 2 subsets of K of size K. The theorem of

Baumgartner (Theorem 6.1.1 a) gives the same conclusion under a weaker hypo-

thesis; in section 6 we give a new proof of that theorem.

Let us call a partial function on K a function whose domain is a sub-

set of K of size K. Two partial functions f and g are almost disjoint

if the set of all a such that f(a) = g(a) has size less than K. Let I

be an ideal over K. A partial function f on K is an I-function if the

domain of f has positive measure. If f and g are I-functions, then

(1.6) f - g mod I

means that {a i dom(f) U dom(g) : f(a) 4 g(a)} € I.

Note that if two I-functions f and g are almost disjoint then f 4 g mod I.

The following lemma establishes a relation between almost disjoint

functions and almost disjoint sets:

Lemma 1.3.1. (a) Let X be a cardinal and let F be a family of size X of

almost disjoint partial functions from K to K. Then there exists a family

of X almost disjoint subsets of K of size K.

(b) Let I be an ideal over K. Let X K be a regular cardinal, let

V K, and let F be a family of size X of almost disjoint I-functions from

K to v. Then there exists a family of X almost disjoint sets of positive

measure.

Proof. (a) If we identify each f € F with its graph, a subset of K

X

K,

we immediately obtain almost disjoint subsets of K X K of size K.

(b) Since I is K-complete, there exists for each f ( : F an ordinal

Y- V such that the set S ={a € dom(f) : f(a) - Yf) has positive measure.

Since X is regular and greater than K, there is a family G c_ F of size