Erdős Problem #792

Definitions and mathematical statement

Shared notation.

Unless a section says otherwise, ${\mathbb{N}}=\{1,2,\ldots\}$, ${\mathbb{Z}}$ is the ring of integers, ${\mathbb{Q}},{\mathbb{R}},{\mathbb{C}}$ are the rational, real, and complex fields, $[N]=\{1,\ldots,\lfloor N\rfloor\}$, and $\log$ is the natural logarithm. For real functions of a parameter tending to infinity, $f\ll g$ and $f=O(g)$ mean $|f|\leq Cg$ eventually for some fixed $C$; $f\gg g$ reverses this comparison; $f=o(g)$ means $f/g\to0$; $f\sim g$ means $f/g\to1$; and $f\asymp g$ means both order bounds. A subscript on $O$ or $\ll$ specifies allowed constant dependence. The expression $n^{o(1)}$ in an upper bound means growth smaller than $n^\epsilon$ for each fixed $\epsilon>0$. Local definitions govern any exception, finite-field convention, or use of the same symbol for another object. An asymptotic claim is not automatically an effective algorithm.

An ordinary sum-free set $B$ satisfies $(B+B)\cap B=\varnothing$, allowing equal summands in $B+B$. In particular it cannot contain zero. The extremal guarantee is $f(n)=\min_{A\subset{\mathbb{Z}},\,|A|=n}\max\{|B|:B\subseteq A,\ (B+B)\cap B=\varnothing\}$. This is different from forbidding subset sums of arbitrary length. [S1] [S2]

Statement recorded in the supplied source.

Let $f(n)$ be maximal such that in any $A\subset \mathbb{Z}$ with $\lvert A\rvert=n$ there exists some sum-free subset $B\subseteq A$ with $\lvert B\rvert \geq f(n)$, so that there are no solutions to $ a+b=c $ with $a,b,c\in B$. Estimate $f(n)$. [S1]

Residual task reported by the archive.

The embedded review (2026-08-17) records: Determine the excess over n/3. [S1]

Short English statement

What is the exact largest sum-free subset size that can be guaranteed in every finite integer set of a prescribed size?

Sources

• S1 ULAM.AI, UnsolvedMath, supplied problems.json, record 2309 (EP-792), statement and background. The embedded literature-review date is 2026-08-17; that is the archive’s review date, not a fresh certification. Dataset landing page: https://huggingface.co/datasets/ulamai/UnsolvedMath.

• S2 T. F. Bloom, Erdős Problems, Problem 792, https://www.erdosproblems.com/792. Reference from the supplied record; the live problem page was not independently checked for this compilation.

• S3 Alon, N. and Kleitman, D. J., Sum-free subsets. (1990), 13--26. Bibliographic information from the supplied record.

• S4 B. Bedert, Large sum-free subsets of sets of integers via $L^1$-estimates for trigonometric sums. arXiv:2502.08624 (2025). Bibliographic information from the supplied record.

• S5 Bourgain, Jean, Estimates related to sumfree subsets of sets of integers. Israel J. Math. (1997), 71-92. Bibliographic information from the supplied record.

• S6 Eberhard, Sean and Green, Ben and Manners, Freddie, Sets of integers with no large sum-free subset. Ann. of Math. (2) (2014), 621-652. Bibliographic information from the supplied record.