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

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 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^[1] that the answer is ${\displaystyle O(2^{N/2})}$, as predicted by the Cameron–Erdős conjecture.^[2]
• How many sum-free sets does an abelian group G contain?^[3]
• What is the size of the largest sum-free set that an abelian group G contains?^[3]

A sum-free set is said to be maximal if it is not a proper subset of another sum-free set.

Let ${\displaystyle f:[1,\infty )\to [1,\infty )}$ be defined by ${\displaystyle f(n)}$ is the largest number ${\displaystyle k}$ such that any subset of ${\displaystyle [1,\infty )}$ with size n has a sum-free subset of size k. The function is subadditive, and by the Fekete subadditivity lemma, ${\displaystyle \lim _{n}{\frac {f(n)}{n}}}$ exists.

Erdős proved that ${\displaystyle \lim _{n}{\frac {f(n)}{n}}\geq {\frac {1}{3}}}$, and conjectured that equality holds.^[4] This was proved in 2014 by Eberhard, Green, and Manners giving an upper bound matching Erdős' lower bound up to a function or order ${\displaystyle o(n)}$, ${\displaystyle f(n)\leq {\frac {n}{3}}+o(n)}$.^[5]

Erdős also asked if ${\displaystyle f(n)\geq {\frac {n}{3}}+\omega (n)}$ for some ${\displaystyle \omega (n)\rightarrow \infty }$, in 2025 Bedert in a preprint proved this giving the lower bound ${\displaystyle f(n)\geq {\frac {n}{3}}+c\log \log n}$^[6][7].

See also

• Erdős–Szemerédi theorem
• Sum-free sequence