Sum-free set

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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:

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


  1. Ben Green, The Cameron–Erdős conjecture, Bulletin of the London Mathematical Society 36 (2004) pp.769-778
  2. P.J. Cameron and P. Erdős, On the number of sets of integers with various properties, Number theory (Banff, 1988), de Gruyter, Berlin 1990, pp.61-79
  3. 3.0 3.1 Ben Green and Imre Ruzsa, Sum-free sets in abelian groups, 2005.