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
a
+
b
=
c
has no solution with
a
,
b
,
c
∈
A
.

For example, the set of odd numbers is a sum-free subset of the integers, and the set *{N/2+1, ..., N}* forms a large sum-free subset of the set *{1,...,N}* (*N* even). Fermat's Last Theorem is the statement that the set of all nonzero *n*^{th} powers is a sum-free subset of the integers for *n* > 2.

Some basic questions that have been asked about sum-free sets are:

How many sum-free subsets of *{1, ..., N}* are there, for an integer *N*? Ben Green has shown that the answer is
O
(
2
N
/
2
)
, as predicted by the Cameron–Erdős conjecture (see Sloane's A007865).
How many sum-free sets does an abelian group *G* contain?
What is the size of the largest sum-free set that an abelian group *G* contains?
A sum-free set is said to be **maximal** if it is not a proper subset of another sum-free set.