Erdős Problem #774

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.

For a dissociated set, distinct finite subsets have distinct sums, including the empty subset. Proportional dissociation uses one constant $c>0$ for all finite $B\subset A$: some dissociated $D\subset B$ has $|D|\geq c|B|$. [S1] [S2]

Statement recorded in the supplied source.

We call $A\subset \mathbb{N}$ dissociated if $\sum_{n\in X}n\neq \sum_{m\in Y}m$ for all finite $X,Y\subset A$ with $X\neq Y$. Let $A\subset \mathbb{N}$ be an infinite set. We call $A$ proportionately dissociated if every finite $B\subset A$ contains a dissociated set of size $\gg \lvert B\rvert$. Is every proportionately dissociated set the union of a finite number of dissociated sets? [S1]

Residual task reported by the archive.

The embedded review (2026-08-17) records: Prove the finite-union conclusion for proportionately dissociated integer sets or construct a genuinely dissociated/free counterexample. [S1]

Transcription note. Only unambiguous escape/formatting damage and nonmathematical serialization debris were repaired in the displayed source statement.

Short English statement

If every finite part of an infinite integer set contains a fixed positive proportion with distinct subset sums, must the whole set split into finitely many such sets?

Sources

• S1 ULAM.AI, UnsolvedMath, supplied problems.json, record 2298 (EP-774), 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 774, https://www.erdosproblems.com/774. Reference from the supplied record; the live problem page was not independently checked for this compilation.

• S3 Gilles Pisier, Arithmetic characterizations of Sidon sets, Bulletin of the American Mathematical Society 8 (1983), 87--89. https://doi.org/10.1090/S0273-0979-1983-15092-9. Bibliographic information from the supplied record.

• S4 Jaroslav Nešetřil, Vojtěch Rödl, and Marcelo Sales, On Pisier Type Theorems, Combinatorica 44 (2024), 1211--1232. https://doi.org/10.1007/s00493-024-00115-1. Bibliographic information from the supplied record.

• S5 Alon, Noga and Erdős, P., An application of graph theory to additive number theory. European J. Combin. (1985), 201-203. Bibliographic information from the supplied record.

• S6 Nešetřil, Jaroslav and R"odl, Vojtěch and Sales, Marcelo, On {P}isier type theorems. Combinatorica (2024), 1211--1232. Bibliographic information from the supplied record.