Sum-free set

{{Short description|Set disjoint from its sumset with itself}}

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 $a,b,c\; \backslash in\; A$.

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{{cite journal |first=Ben |last=Green |arxiv=math.NT/0304058 |title=The Cameron–Erdős conjecture |journal=Bulletin of the London Mathematical Society |volume=36 |issue=6 |date=November 2004 |pages=769–778 | doi=10.1112/S0024609304003650 |mr=2083752}} that the answer is $O(2^\{N/2\})$, 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 {{OEIS2C|id=A007865}}
• How many sum-free sets does an abelian group G contain?Ben Green and Imre Ruzsa, [http://www.arxiv.org/pdf/math.CO/0307142 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 $f:\; [1,\; \backslash infty)\; \backslash to\; [1,\; \backslash infty)$ be defined by $f(n)$ is the largest number $k$ such that any subset of $[1,\; \backslash infty)$ with size n has a sum-free subset of size k. The function is subadditive, and by the Fekete subadditivity lemma, $\backslash lim\_n\backslash frac\{f(n)\}\{n\}$ exists.

Erdős proved that $\backslash lim\_n\backslash frac\{f(n)\}\{n\}\; \backslash geq\; \backslash frac\; 13$, 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 $o(n)$, $f(n)\; \backslash leq\; \backslash frac\{n\}\{3\}+o(n)$.{{Cite journal |last1=Eberhard |first1=Sean |last2=Green |first2=Ben |last3=Manners |first3=Freddie |date=2014 |title=Sets of integers with no large sum-free subset |url=https://www.jstor.org/stable/24522935 |journal=Annals of Mathematics |volume=180 |issue=2 |pages=621–652 |doi=10.4007/annals.2014.180.2.5 |jstor=24522935 |issn=0003-486X|arxiv=1301.4579 }}

Erdős also asked if $f(n)\backslash geq\; \backslash frac\{n\}\{3\}+\backslash omega(n)$ for some $\backslash omega(n)\backslash rightarrow\; \backslash infty$, in 2025 Bedert in a preprint proved this giving the lower bound $f(n)\backslash geq\; \backslash frac\{n\}\{3\}+c\backslash log\backslash log\; n$.{{cite arXiv | eprint=2502.08624 | last1=Bedert | first1=Benjamin | title=Large sum-free subsets of sets of integers via L¹-estimates for trigonometric series | date=2025 | class=math.NT }}{{Cite web |last=Sloman |first=Leila |date=2025-05-22 |title=Graduate Student Solves Classic Problem About the Limits of Addition |url=https://www.quantamagazine.org/graduate-student-solves-classic-problem-about-the-limits-of-addition-20250522/ |access-date=2025-05-23 |website=Quanta Magazine |language=en}}