Set mappings defined on pairs

Abstract

A set mapping on pairs over the set S is a function/ such that for each unordered pair a of elements of S,f(a) is a subset of 5 disjoint from a. A subset H of S is said to be free for/if x £ f{{y, z}) ' o r all x, y, z from H. In this paper, we investigate conditions imposed on the range of/which ensure that there is a large set free for/. For example, we show that if/is defined on a set of size K ++ with always \f(a)\ < K then/has a free set of size K + if the range of/satisfies the »c-chain condition, or if any two sets in the range of / have an intersection of size less than 9 for some 9 with 9 < K.