Sum-free set

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 $a,b,c \in A$.

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^[1] that the answer is $O(2^{N/2})$, as predicted by the Cameron–Erdős conjecture^[2] (see Sloane's  A007865).
• How many sum-free sets does an abelian group G contain?^[3]
• What is the size of the largest sum-free set that an abelian group G contains?^[3]

A sum-free set is said to be maximal if it is not a proper subset of another sum-free set.

References

1. Ben Green, The Cameron–Erdős conjecture, Bulletin of the London Mathematical Society 36 (2004) pp.769-778
2. P.J. Cameron and P. Erdős, On the number of sets of integers with various properties, Number theory (Banff, 1988), de Gruyter, Berlin 1990, pp.61-79
3. Ben Green and Imre Ruzsa, Sum-free sets in abelian groups, 2005.