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 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 ''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 are there, for an integer ''N''? Ben Green has shown that the answer is $O(2^)$, as predicted by the Cameron–Erdős conjecture.
* How many sum-free sets does an abelian group ''G'' contain?Ben Green and Imre Ruzsa
Sum-free sets in abelian groups
2005.
* 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 $f:$function is subadditive, and by the Subadditivity#Properties, Fekete subadditivity lemma, $\lim_n\frac$ exists.

Erdős mathematical proof, proved that $\lim_n\frac \geq \frac 13$, 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 $o(n)$, $f(n) \leq \frac+o(n)$.

Erdős also asked if $f(n)\geq \frac+\omega(n)$ for some $\omega(n)\rightarrow \infty$, in 2025 Bedert in a preprint proved this giving the lower bound $f(n)\geq \frac+c\log\log n$.

See also

* Erdős–Szemerédi theorem
* Sum-free sequence

References

{{reflist

External links

*
Erdős Problem 792
- erdosproblems.com

Sumsets
Additive combinatorics