Sum-free set

of an Abelian group or semigroup $M$

A subset $S \subset M$ with the property that no element of $S$ is equal to a sum of two elements of $S$. It is known that in general a subset of $M$ of size $n$ contains a sum-free subset of order $> \frac27 n$. The set of odd numbers in $\{1,\ldots,n\}$ is a sum-free set of order $\frac12 n$.

The Cameron–Erdős conjecture, proved by Ben Green in 2003, asserts that the set $\{1,\ldots,n\}$ contains $O\left({2^{n/2}}\right)$ sum-free sets.

References

• Guy, Richard K. Unsolved problems in number theory (3rd ed.) Springer-Verlag (2004) ISBN 0-387-20860-7 Zbl 1058.11001