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
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