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 has no solution with .
For example, the set of odd numbers is a sum-free subset of the integers, and the set {N/2+1, ..., N} forms a large sum-free subset of the set {1,...,N} (N even). Fermat's Last Theorem is the statement that the set of all nonzero nth powers is a sum-free subset of the integers for n > 2.
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, as predicted by the Cameron–Erdős conjecture (see Sloane's A007865).
- 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.
Famous quotes containing the word set:
“The extra worry began it—on the
Blue blue mountain—she never set foot
And then and there. Meanwhile the host
Mourned her quiet tenure. They all stayed chatting.
No one did much about eating.”
—John Ashbery (b. 1927)