Large Sum-Free Sets

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

A subset $B\subset\mathbb Z$ is sum-free if there are no $x,y,z\in B$ with $x+y=z$; the summands $x,y$ need not be distinct. For each finite $A\subset\mathbb N$, write \[ s(A)=\max\{|B|:B\subset A\text{ is sum-free}\},\qquad s(n)=\min_{\substack{A\subset\mathbb N\\|A|=n}}s(A). \] The catalogue's symbol for the secondary term denotes a function tending to infinity, rather than the usual asymptotic lower-bound notation.

Statement recorded in the supplied source.

Does there exist a function $\omega:\mathbb N\to\mathbb R$ with $\omega(n)\to+\infty$ as $n\to\infty$ such that every $n$-element set $A\subset\mathbb N$ contains a sum-free subset $B\subset A$ satisfying \[ |B|\ge \frac n3+\omega(n) \] for all sufficiently large $n$? Equivalently, is $s(n)-n/3\to+\infty$? [S1][S2] Bedert's Theorem 1.2 answers this historical question affirmatively by proving an additive improvement $c\log\log n$ for some absolute $c>0$. Here $\log$ denotes the natural logarithm, and the estimate is interpreted for sufficiently large $n$. [S2]

Short English statement

Must every set of $n$ positive integers contain a sum-free subset whose size exceeds $n/3$ by an amount tending to infinity with $n$? Bedert's 2025 theorem gives an affirmative answer.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 92 (GREEN-001). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Benjamin Bedert, Large sum-free subsets of sets of integers via $L^1$-estimates for trigonometric series, arXiv:2502.08624v1, 12 February 2025, section 1, definition (1), Problem 1.1 and Theorem 1.2, pp. 1--2; Problem 1.1 explicitly identifies Green's Problem 1. https://arxiv.org/pdf/2502.08624v1.
• S3 Ben Green, 100 Open Problems, author manuscript with December 2025 updates, Problem 1, pp. 2--3. https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf.