Restricted Sumset Problem

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.

For a finite set $S\subset\mathbb Z$, its restricted sumset is \[ S\widehat{+}S=\{s+t:s,t\in S,\ s\ne t\}. \] Thus the prohibition concerns sums of distinct elements only; a double $2s$ is permitted to belong to the ambient set. Throughout this statement $\log$ is the natural logarithm.

Statement recorded in the supplied source.

Is there $n_0\in\mathbb N$ such that, for every integer $n\ge n_0$ and every set $A\subset\mathbb Z$ with $|A|=n$, there exists $S\subset A$ for which \[ |S|\ge(\log n)^{100},\qquad (S\widehat{+}S)\cap A=\varnothing? \] Equivalently, must the largest subset sum-free with respect to $A$ have at least $(\log |A|)^{100}$ elements uniformly over all sufficiently large finite integer sets? [S1][S2]

Short English statement

Can every sufficiently large finite set of integers provide a subset of size at least the hundredth power of its logarithmic size such that no sum of two distinct chosen elements belongs to the original set?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 93 (GREEN-002). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Adrian Beker, The Erdős--Moser sum-free set problem via improved bounds for $k$-configurations, arXiv:2501.10203v2, 2 October 2026, section 1, pp. 1--3: sum-free-with-respect-to-$A$ definition and explicit identification of Green's Problem 2. The catalogue specifies the target exponent $100$; the paper's proved lower bound in Theorem 1.2 has exponent $1+c$. https://arxiv.org/pdf/2501.10203v2.
• S3 Ben Green, 100 Open Problems, author manuscript with December 2025 updates, Problem 2, pp. 3--4. https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf.