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.

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 set.

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, as predicted by the Cameron–Erdős conjecture.

• 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.

such that any set of n nonzero integers has a sum-free subset of size k. The function is subadditive, and by the Fekete subadditivity lemma exists.

Erdős proved, and conjectured that equality holds..

.

See also

• Erdős–Szemerédi theorem

• Sum-free sequence

References

External links

• Erdős Problem 792 - erdosproblems.com