Problem 298

Does every set $A\subseteq \mathbb{N}$ of positive density contain some finite $S\subset A$ such that $\sum_{n\in S}\frac{1}{n}=1$ ?

Source: erdosproblems.com/298

Number theory
Unit fractions

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SolvedProved

An accepted solution exists. The statement is true.

Proved, in the site's label. Bloom's theorem applies under the assumption of positive upper density, so it answers the Statement in each of its readings. The site's label is PROVED (LEAN); the existing formal proof and statement-only declarations are distinguished below.

ErGr80: P. Erdős and R. L. Graham, Old and new problems and results in combinatorial number theory, Monographies de L'Enseignement Mathématique (1980). Bloom's paper locates the density question on p. 36.
Er92c: P. Erdős, Some of my forgotten problems in number theory, Hardy–Ramanujan Journal 15 (1992), 34–50. Original reference listed by the site.
Bl21: T. F. Bloom, On a density conjecture about unit fractions, arXiv:2112.03726 (2021), v2 (2023), with Appendix B co-written by T. F. Bloom and B. Mehta; JEMS 27 (2025), 4563–4589.
LiSa24: Y. P. Liu and M. Sawhney, On further questions regarding unit fractions, arXiv:2404.07113v1 (10 April 2024), Theorem 1.1, p. 1; proof pp. 19–20.

Problem 298

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