On the structure of a random sum-free set of positive integers

Cameron introduced a natural probability measure on the set S of sum-free sets, and asked which sets of sum-free sets have a positive probability of occurring in this probability measure. He showed that the set of subsets of the odd numbers has a positive probability, and that the set of subsets of any sum-free set corresponding to a complete modular sum-free set also has a positive probability of occurring. In this paper we consider, for every sum-free set S, the representation function r(S)(n), and show that if r(S)(n) grows sufficiently quickly then the set of subsets of S has positive probability, and conversely, that if r(S)(n) has a sub-sequence with suitably slow growth, then the set of subsets of S has probability zero. The results include those of Cameron mentioned above as particular cases. (C) 1998 Elsevier Science B.V. All rights reserved.