Erdős Problem #949

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.

The continuum is the cardinality $|{\mathbb{R}}|=2^{\aleph_0}$. Both $A$ and $A+A$ must lie in the complement of the same given sum-free set $S$. Equal summands are allowed in $A+A$. No measurability of $S$ or $A$ is assumed. [S1] [S2]

Statement recorded in the supplied source.

Let $S\subset \mathbb{R}$ be a set containing no solutions to $a+b=c$. Must there be a set $A\subseteq \mathbb{R}\backslash S$ of cardinality continuum such that $A+A\subseteq \mathbb{R}\backslash S$? [S1]

Residual task reported by the archive.

The embedded review (2026-08-17) records: Remove the Sidon assumption or build a counterexample. [S1]

Short English statement

Outside any sum-free subset of the reals, can one find continuum many elements whose pairwise sums also avoid that original subset?

Sources

• S1 ULAM.AI, UnsolvedMath, supplied problems.json, record 2401 (EP-949), 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 949, https://www.erdosproblems.com/949. Reference from the supplied record; the live problem page was not independently checked for this compilation.