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 has no solution with .
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 nth 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, 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.
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