Problem 300

Let $A(N)$ denote the maximal cardinality of $A\subseteq \{1,\ldots,N\}$ such that $\sum_{n\in S}\frac{1}{n}\neq 1$ for all $S\subseteq A$ . Estimate $A(N)$ .

Source: erdosproblems.com/300

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.

Solved, in the site's label, which marks an estimate carried out rather than a proof or disproof: $A(N)=(1-1/e+o(1))N$ , by Liu and Sawhney's Theorem 1.3 (Int. Math. Res. Not. 2026) together with the trivial lower bound. Erdős and Graham had expected $A(N)=(1+o(1))N$ ; the site credits Croot's 2003 work with the first disproof, $A(N)<cN$ for some $c<1$ . The site's label is SOLVED (LEAN); the Lean behind the suffix is a file in Boris Alexeev's lean-proofs collection that declares itself a formalization of Liu and Sawhney's theorem, with Codex and GPT-5.6 Sol as formal authors, linked on their claim page and described under Existing formalization. The corpus has not built it and claims no formalized evidence.

LiSa24: Liu, Y. P. and Sawhney, M., On further questions regarding unit fractions. arXiv:2404.07113v1 (10 April 2024); Int. Math. Res. Not. 2026, no. 2, rnaf382, DOI 10.1093/imrn/rnaf382, published online 14 January 2026. Theorem 1.3. Library home: Liu and Sawhney 2024.
Cr03: Croot, III, Ernest S., On a coloring conjecture about unit fractions. Ann. of Math. (2) 157 (2003), no. 2, 545–556; arXiv:math/0311421. Library home: Croot 2003.
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.
Va99: Various, Some of Paul's favorite problems. Booklet produced for the conference "Paul Erdős and his mathematics", Budapest, July 1999; item 1.14 as cited by the site. Library home: Various 1999; item 1.14 (printed p. 3), whose first line the scan cuts at the margin after " $\le n$ ": "What is the maximum number of integers $a_1<a_2<\dots<a_k\le n$ [such] that no sum $\sum\frac{\epsilon_i}{a_i}$ , ( $\epsilon_i=0,1$ ) equals 1? Can $k$ be $n-o(n)$ ?"

Problem 300

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