Large sum-free sets in ternary spaces

AbstractWe show that any sum-free subset A⊆F3r (with an integer r⩾3), satisfying |A|>5·3r-3, is contained in a hyperplane. This bound is best possible: there exist sum-free subsets of cardinality |A|=5·3r-3 not contained in a hyperplane.Conjecturally, if A⊆F3r is maximal sum-free and aperiodic (not a union of cosets of a non-zero subgroup), then |A|⩽(3r-1+1)/2. If true, this is best possible and allows one for any fixed ε>0 to establish the structure of all sum-free subsets A⊆F3r such that |A|>(1/6+ε)·3r