Problem 168

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claims/: The 1 claim page of Problem 168, one per claimant's result; the problem's standing derives from them.

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Statement. Let $F(N)$ be the size of the largest subset of $\{1,\ldots,N\}$ which does not contain any set of the form $\{n,2n,3n\}$. What is

$$\lim_{N\to \infty}\frac{F(N)}{N}?$$

Is this limit irrational?

Status. Open. The site's label is OPEN (page last edited 23 March 2026). One partial claim is recorded: Graham, Witsenhausen and Spencer proved that the limit exists and equals $\frac13\sum_{k\in K}1/d_k$, a series over the $3$-smooth numbers $d_1<d_2<\cdots$ indexed by the set $K$ of $k$ at which the extremal count on $\{d_1,\ldots,d_k\}$ grows; the site's commentary credits the result and reports Eberhard's evaluation of the series as $0.800965\cdots$. The paper appeared in a collected volume not shown to be refereed, so the claim is pending. No closed form for the value and no answer to the irrationality question is claimed, so the problem stays open.

Source. erdosproblems.com/168, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #168, https://www.erdosproblems.com/168.

References.

• [GSW77] Graham, R. and Spencer, J. and Witsenhausen, H., On Extremal Density Theorems for Linear Forms. Number Theory and Algebra (1977).

Formalization. Statement in formal-conjectures.

Progress

Not yet compiled.

Known Results

Not yet compiled.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• erdos_1979_old_new_problems_results_combinatorial_number
• graham_1977_extremal_density_theorems_linear_forms
• graham_1977_extremal_density_theorems_linear_forms / equation_12
• graham_1977_extremal_density_theorems_linear_forms / theorem_2