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 has no solution with.
For example, the set of odd numbers is a sum-free subset of the integers, and the set forms a large sum-free subset of the set. 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 are there, for an integer N? Ben Green has shown that the answer is, 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?
Let be defined by is the largest number 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 that, and conjectured that equality holds. This was proved in 2014 by Eberhard, Green, and Manners giving an upper bound matching Erdős' lower bound up to a function or order, .
Erdős also asked if for some, in 2025 Bedert in a preprint proved this giving the lower bound.