Sum-free set (nonfiction)

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 {\displaystyle a+b=c}a+b=c has no solution with {\displaystyle a,b,c\in A}{\displaystyle a,b,c\in A}.

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 subset.

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 {\displaystyle O(2^{N/2})}{\displaystyle O(2^{N/2})}, as predicted by the Cameron–Erdős conjecture[2] (see Sloane's OEIS: 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.