A solution to a problem of Cameron on sum-free complete sets

For any subsets A and B of an additive group G, define A + B = { a + b: a ϵ A and b ϵ B} and −A = {−a: a ϵ A}. A subset S of G is said to be sum-free, complete, and symmetric respectively if S + S ⊂ Sc, S + S ⊃ Sc, and S = −S. Cameron asked if for all sufficiently large moduli m there exists a sum-free complete set in ZmZ that is not symmetric. We answer Cameronʹs question by showing there exists such a set for all moduli greater than or equal to 890626. We also show that every sum-free complete set in ZmZ that is not symmetric can be used to construct a counter-example to a conjecture of Conway disproved by Marica. Conway conjectured that for any finite set S of integers, |S + S| ⩽ |S  S|.