Sum-free set

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 + 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 set.

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.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; see Sloane How many sum-free sets does an abelian group G contain?Ben Green and Imre Ruzsa, Sum-free sets in abelian groups, 2005. 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.

Let be defined by is the largest number such that any set of n nonzero integers has a sum-free subset of size k. The function is subadditive, and by the Fekete subadditivity lemma, exists.

Erdős proved that , and conjectured that equality holds.P. Erdős, "Extremal problems in number theory", Matematika, 11:2 (1967), 98–105; Proc. Sympos. Pure Math., Vol. VIII, 1965, 181–189 This was proved in 2014 by Eberhard, Green, and Manners giving an upper bound matching Erdős' lower bound up to a function or order , .

Erdős also asked if for some , in 2025 Bedert in a preprint proved this giving the lower bound .

Sum-free set

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