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 AA 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+1, ..., 2N} forms a large sum-free subset of the set {1,...,2N}. Fermat's Last Theorem is the statement that, for a given integer n > 2, the set of all nonzero nth powers of the integers is a sum-free subset.

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[1] that the answer is , as predicted by the Cameron–Erdős conjecture[2] (see Sloane's OEISA007865).
  • How many sum-free sets does an abelian group G contain?[3]
  • What is the size of the largest sum-free set that an abelian group G contains?[3]

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

Let be defined by is the largest number such that any subset of with size n has a sum-free subset of size k. The function is subadditive, and by Fekete subadditivity lemma, exists. Erdos proved that , and conjectured that it is exact("P. Erdős, "Extremal problems in number theory", Matematika, 11:2 (1967), 98–105; Proc. Sympos. Pure Math., Vol. VIII, 1965, 181–189"). This is proved in.[4]

References[edit]

  1. ^ Green, Ben (November 2004). "The Cameron–Erdős conjecture". Bulletin of the London Mathematical Society. 36 (6): 769–778. arXiv:math.NT/0304058. doi:10.1112/S0024609304003650. MR 2083752.
  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. ^ a b Ben Green and Imre Ruzsa, Sum-free sets in abelian groups, 2005.
  4. ^ Eberhard, Sean; Green, Ben; Manners, Freddie (2014). "Sets of integers with no large sum-free subset". Annals of Mathematics. 180 (2): 621–652. ISSN 0003-486X.