Problem 296

Let $N\geq 1$ and let $k(N)$ be maximal such that there are $k$ disjoint $A_1,\ldots,A_k\subseteq \{1,\ldots,N\}$ with $\sum_{n\in A_i}\frac{1}{n}=1$ for all $i$ . Estimate $k(N)$ . Is it true that $k(N)=o(\log N)$ ?

Source: erdosproblems.com/296

Number theory
Unit fractions

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An accepted solution exists. Settled in another form, for example when its parts resolve differently or the question is open-ended.

PROVED (LEAN) is the site's label, which attaches to the estimate: $k(N)=(1-o(1))\log N$ , by Bloom's Theorem 3 (J. Eur. Math. Soc. 27 (2025)) combined with a greedy removal argument, an observation the site credits to Hunter and Sawhney and which is written out below. This is the accepted claim Hunter and Sawhney, recorded with the value answered: it determines $k(N)$ to first order and answers the closing question in the negative, since $k(N)/\log N\to1$ rules out $k(N)=o(\log N)$ . The Lean suffix is a catalog label qualified under Existing formalization below; the Lean files are formalization links on the claim page, and no local kernel credit is claimed.

ErGr80: Erdős, P. and Graham, R. L., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathématique 28, Université de Genève (1980), printed p. 36. Library home: Erdős and Graham 1980.
Bl21: Bloom, T. F., On a density conjecture about unit fractions, with an appendix co-written by Bloom and B. Mehta. arXiv:2112.03726 (2021), v2 (12 October 2023); J. Eur. Math. Soc. 27 (2025), no. 11, 4563--4589, DOI 10.4171/JEMS/1456. Theorem 3 (Theorem 1.3 in the published version). Library home: Bloom 2021.
LiSa24: Liu, Y. P. and Sawhney, M., On further questions regarding unit fractions. arXiv:2404.07113v1 (2024); Int. Math. Res. Not. 2026, no. 2, rnaf382. Theorem 1.1, which sharpens the error term below. Library home: Liu and Sawhney 2024.

Problem 296

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