# 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 AA 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+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 subset.

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 $O(2^{N/2})$ , as predicted by the Cameron–Erdős conjecture (see Sloane's ).
• 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 Fekete subadditivity lemma, $\lim _{n}{\frac {f(n)}{n}}$ exists. Erdos proved that $\lim _{n}{\frac {f(n)}{n}}\geq {\frac {1}{3}}$ , and conjectured that it is exact("P. Erdős, "Extremal problems in number theory", Matematika, 11:2 (1967), 98–105; Proc. Sympos. Pure Math., Vol. VIII, 1965, 181–189"). This is proved in.