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 $$a + b = c$$ has no solution with $$a,b,c \in A$$.

For example, the set of odd numbers is a sum-free subset of the integers, and the set {N +&thinsp;1, ..., 2N&hairsp;} forms a large sum-free subset of the set {1, ..., 2N&hairsp;}. 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&hairsp;} are there, for an integer N? Ben Green has shown that the answer is $$O(2^{N/2})$$, 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.

Let $$f: [1, \infty) \to [1, \infty)$$ be defined by $$f(n)$$ is the largest number $$k$$ such that any subset of $$[1, \infty)$$ with size n has a sum-free subset of size k. The function is subadditive, and by the Fekete subadditivity lemma, $$\lim_n\frac{f(n)}{n}$$ exists. Erdős proved that $$\lim_n\frac{f(n)}{n} \geq \frac 13$$, and conjectured that equality holds. This was proved by Eberhard, Green, and Manners.