# Sum-free set

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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 *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 , 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

- ↑ Ben Green,
*The Cameron–Erdős conjecture*, Bulletin of the London Mathematical Society**36**(2004) pp.769-778 - ↑ 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 - ↑
^{3.0}^{3.1}Ben Green and Imre Ruzsa, Sum-free sets in abelian groups, 2005.