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 [math]\displaystyle{ a + b = c }[/math] has no solution with [math]\displaystyle{ a,b,c \in A }[/math]. 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 n^th 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 that the answer is [math]\displaystyle{ O(2^{N/2}) }[/math], 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 [math]\displaystyle{ f: [1, \infty) \to [1, \infty) }[/math] be defined by [math]\displaystyle{ f(n) }[/math] is the largest number [math]\displaystyle{ k }[/math] such that any subset of [math]\displaystyle{ [1, \infty) }[/math] with size n has a sum-free subset of size k. The function is subadditive, and by the Fekete subadditivity lemma, [math]\displaystyle{ \lim_n\frac{f(n)}{n} }[/math] exists. Erdős proved that [math]\displaystyle{ \lim_n\frac{f(n)}{n} \geq \frac 13 }[/math], and conjectured that equality holds. This was proved by Eberhard, Green, and Manners.

See also

• Erdős–Szemerédi theorem
• Sum-free sequence