In additive combinatorics and number theory, a subset A of an abelian group G is said to be sum-free if the sumset A + A is disjoint from A.
In other words, A is sum-free if the equation
For example, the set of odd numbers is a sum-free subset of the integers, and the set {N + 1, ..., 2N } forms a large sum-free subset of the set {1, ..., 2N }.
Fermat's Last Theorem is the statement that, for a given integer n > 2, the set of all nonzero nth powers of the integers is a sum-free set.
Some basic questions that have been asked about sum-free sets are: A sum-free set is said to be maximal if it is not a proper subset of another sum-free set.
with size n has a sum-free subset of size k. The function is subadditive, and by the Fekete subadditivity lemma,
, and conjectured that equality holds.
[4] This was proved by Eberhard, Green, and Manners.