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Sum-free set

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of an Abelian group or semigroup

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
How to Cite This Entry:
Sum-free set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Sum-free_set&oldid=54611