Large sum-free sets in ternary spaces

We show that any sum-free subset A subset of F-3(r) (with an integer r >= 3), satisfying vertical bar A vertical bar > 5 (.) 3(r-3), is contained in a hyperplane. This bound is best possible: there exist sum-free subsets of cardinality vertical bar A vertical bar = 5 (.) 3(r-3) not contained in a hyperplane. Conjecturally, if A subset of F-3(r) is maximal sum-free and aperiodic (not a union of cosets of a non-zero subgroup), then vertical bar A broken vertical bar <= (3(r-1) +])/2. If true, this is best possible and allows one for any fixed epsilon > 0 to establish the structure of all sum-free subsets A subset of F-3(r) such that vertical bar A vertical bar > (1/6 + epsilon) (.) 3(r). (c) 2005 Elsevier Inc. All rights reserved.