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 ${\displaystyle a+b=c}$ has no solution with ${\displaystyle a,b,c\in A}$.

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 n^th 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[1] that the answer is ${\displaystyle O(2^{N/2})}$, as predicted by the Cameron–Erdős conjecture.[2]
• 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 ${\displaystyle f:[1,\infty )\to [1,\infty )}$ be defined by ${\displaystyle f(n)}$ is the largest number ${\displaystyle k}$ such that any subset of ${\displaystyle [1,\infty )}$ with size n has a sum-free subset of size k. The function is subadditive, and by the Fekete subadditivity lemma, ${\displaystyle \lim _{n}{\frac {f(n)}{n}}}$ exists. Erdős proved that ${\displaystyle \lim _{n}{\frac {f(n)}{n}}\geq {\frac {1}{3}}}$, and conjectured that equality holds.[4] This was proved by Eberhard, Green, and Manners.[5]

See also

• Erdős–Szemerédi theorem
• Sum-free sequence

References

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; see Sloane OEIS: A007865
3. Ben Green and Imre Ruzsa, Sum-free sets in abelian groups, 2005.
4. P. Erdős, "Extremal problems in number theory", Matematika, 11:2 (1967), 98–105; Proc. Sympos. Pure Math., Vol. VIII, 1965, 181–189
5. Eberhard, Sean; Green, Ben; Manners, Freddie (2014). "Sets of integers with no large sum-free subset". Annals of Mathematics. 180 (2): 621–652. arXiv:1301.4579. doi:10.4007/annals.2014.180.2.5. ISSN 0003-486X. JSTOR 24522935.