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 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 {N/2+1, ..., N} forms a large sum-free subset of the set {1,...,N} (N even).

Some basic questions that have been asked about sum-free sets are:

  1. How many sum-free subsets of {1, ..., N} are there, for an integer N? The answer has been shown[1][2] to be O(2N / 2), as predicted by the Cameron–Erdős conjecture.[3]
  2. How many sum-free sets does an abelian group G contain?[4]
  3. What is the size of the largest sum-free set that an abelian group G contains?[4]

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

[edit] References

  1. ^ Ben Green, The Cameron–Erdős conjecture, 2003.
  2. ^ Ben Green, The Cameron–Erdős conjecture, Bulletin of the London Mathematical Society 36 (2004) pp.769-778
  3. ^ 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
  4. ^ a b Ben Green and Imre Ruzsa, Sum-free sets in abelian groups, 2005.