Erdős Problem #635

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.

A unit fraction is the reciprocal $1/m$ of a positive integer. An expansion with distinct denominators uses each denominator at most once; reordering its terms does not produce a different underlying denominator set. [S1] [S2]

Statement recorded in the supplied source.

Let $t\geq 1$ and $A\subseteq \{1,\ldots,N\}$ be such that whenever $a,b\in A$ with $b-a\geq t$ we have $b-a\nmid b$. How large can $\lvert A\rvert$ be? Is it true that $ \lvert A\rvert \leq \left(\frac{1}{2}+o_t(1)\right)N? $ [S1]

Residual task reported by the archive.

The embedded review (2026-08-17) records: Determine the true excess above N/2 for fixed t, especially t=2, rather than only an o(N) error. [S1]

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

Short English statement

If sufficiently large differences between chosen integers never divide the larger integer, can the chosen set exceed half the interval asymptotically?

Sources

• S1 ULAM.AI, UnsolvedMath, supplied problems.json, record 2225 (EP-635), 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 635, https://www.erdosproblems.com/635. Reference from the supplied record; the live problem page was not independently checked for this compilation.