Problem 300

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claims/: The 2 claim pages of Problem 300, one per claimant's result; the problem's standing derives from them.

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Statement. 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 $A(N)$.

Formulation. The site's wording as of 2026-09-17 (page last edited 23 January 2026). $A(N)$ is the largest size of a subset of $\{1,\ldots,N\}$ none of whose subsets has reciprocal sum one. For any fixed $\gamma>0$ and large $N$ the integers in $((1/e+\gamma)N,N]$ have reciprocal sum below one, so $A(N)\ge(1-1/e-\gamma)N-1$; the site calls $A(N)\ge(1-1/e+o(1))N$ trivial.

Status. 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.

Source. erdosproblems.com/300, accessed 2026-09-17: the problem page (SOLVED (LEAN); last edited 23 January 2026; OEIS A390393 linked), its empty discussion thread and its empty proof-claim tab. The site cites [ErGr80] and [Va99, 1.14] as the problem's sources and [Cr03] and [LiSa24] in its commentary. Cite as: T. F. Bloom, Erdős Problem #300, https://www.erdosproblems.com/300, accessed 2026-09-17.

References.

• [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_2024_further_questions_regarding_unit_fractions.
• [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_coloring_conjecture_about_unit_fractions.
• [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: erdos_1980_old_new_problems_results_combinatorial_number_theory.
• [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_some_pauls_favorite_problems; 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 $n-o(n)$?"

Formalization. The formal-conjectures statement file for the problem is ErdosProblems/300.lean; it and the Lean 4 proof in Boris Alexeev's lean-proofs collection are described under Existing formalization. The corpus has built neither.

Current assessment

Claims. One accepted full claim settles the problem: Liu and Sawhney's Theorem 1.3 with the trivial lower bound, refereed in Int. Math. Res. Not. 2026 and credited by the site's curator independently of the authors. The disproof of the expected asymptotic that the site credits to Croot is the accepted partial claim Croot's bound, recorded as the attribution the site and Liu and Sawhney make. The frontmatter standing is derived from these pages.

The question. The site asks for the order of $A(N)$, shows SOLVED (LEAN), and says in its commentary that Erdős and Graham believed $A(N)=(1+o(1))N$, that Croot disproved this by showing $A(N)<cN$ for some constant $c<1$ and all large $N$, that $A(N)\ge(1-1/e+o(1))N$ is trivial, and that Liu and Sawhney proved $A(N)=(1-1/e+o(1))N$. It links OEIS A390393. As of 2026-09-17 the thread and the proof-claim tab were empty. The community database lists the problem as solved (Lean), with a last update of 24 August 2026, and its statement as formalized, with a last update of 20 September 2026.

Status support. The status-defining source is Liu and Sawhney's Theorem 1.3, arXiv:2404.07113v1, p. 2: fix $\varepsilon\in(0,1/2)$; for $N$ sufficiently large in terms of $\varepsilon$, every $A\subseteq[1,N]$ with $|A|\ge(1-1/e+\varepsilon)N$ has a subset $A'$ with $\sum_{n\in A'}1/n=1$. The remark following it gives the sharpness: for $\gamma>0$ and $N$ large, $\sum_{n=\lfloor(1/e+\gamma)N\rfloor}^{N}1/n\le\int_{1/e+\gamma/2}^1dx/x<1$, so the top segment has no unit subsum. Together these give $A(N)=(1-1/e+o(1))N$. Acceptance: the paper appeared in Int. Math. Res. Not. 2026, no. 2, rnaf382 (received 28 October 2025, accepted 23 December 2025, online 14 January 2026, per the publisher's record); the locators are those of arXiv v1, and the published text has not been compared. The proof (p. 20) proceeds as follows: discard the integers below $\varepsilon N/2$, so that the remaining reciprocal sum exceeds one by a fixed amount; remove non-smooth integers and integers with many prime factors; prune with the paper's Lemma 6.2; apply its Proposition 5.2 with target one. The theorem page records two parameter conditions of Proposition 5.2 that the printed proof does not visibly meet, and the published version has not been compared on them; the library's coverage of this theorem is statement and sketch, and the status rests on the refereed publication.

The disproof of $A(N)=(1+o(1))N$ predates the asymptotic. Liu and Sawhney write on p. 2: "From work of Croot [7], it follows that if $|A|\geq(1-\delta)N$ for $\delta$ sufficiently small there exists such a subset." Croot's paper (Main Theorem) proves a unit-subsum criterion for heavy sets of smooth integers and does not state this consequence, and neither paper writes out the deduction. It is recorded here as the attribution the site and Liu and Sawhney make, not as a compiled result. The monograph's expectation is on printed p. 36: "Let $A(n)$ denote the largest value of $|S|$ such that $S\subseteq\{1,2,\ldots,n\}$ contains no set in $\mathscr X$. Probably $A(n)=n+o(n)$ but we cannot prove this."

Data lead, not status. OEIS A390393 (H. Raza, 4 November 2025) lists $a(n)$, the maximum size of a subset of $\{1,\ldots,n\}$ with no unit subsum, beginning $0,1,2,3,4,4,5,6,7,8,9,10,11,12,12,\ldots$, and cites the site and the Liu–Sawhney asymptotic. Its terms were not verified here.

Search scope. The search covered the site's problem, discussion and proof-claim pages; the community database record; the formal-conjectures tree, which then had no file for Problem 300; the directory src/v4.29.1/ErdosProblems of Boris Alexeev's lean-proofs collection, which has no file for the problem; the file under src/latest, posted 17 August 2026, is recorded under Existing formalization; the arXiv listings for 2404.07113 (v1 only) and math/0311421 (one version); the Oxford Academic and Annals article records; OEIS A390393; the Semantic Scholar citing-paper records for the Liu–Sawhney and Croot papers (five and twenty records; the 2025 and 2026 items concern approximate reciprocal subsums, partitions with prescribed reciprocal sums, best underapproximations, the count of unit-sum subsets, faithful decompositions of rationals and Rado numbers, none this problem); the arXiv API listing of the sixty most recent abstracts mentioning unit or Egyptian fractions (to 7 September 2026); and two general web searches. MathSciNet, zbMATH, full-text scholarly search engines and X were not searched. Nothing found refines the $o(N)$ term.

Remaining gaps. The proof of Theorem 1.3 is not compiled beyond a sketch and carries two recorded parameter questions; the published text is uncompared; the deduction of $A(N)<cN$ from Croot's theorem is unwritten; the Lean 4 proof behind the label's suffix is unbuilt and unaudited here.

Progress and known results

• Erdős and Graham (1980, printed p. 36): the expectation $A(n)=n+o(n)$.
• Croot (2003): a constant $c<1$ with $A(N)<cN$ for large $N$, as attributed by the site to his paper and by Liu and Sawhney to his work; neither names a theorem or writes the deduction, and the paper's Main Theorem does not state the bound.
• Trivial lower bound: $A(N)\ge(1-1/e-o(1))N$ from the top segment.
• Liu and Sawhney's Theorem 1.3 (2024; published 2026): $A(N)=(1-1/e+o(1))N$. Related: the reciprocal-mass threshold of Problem 47, the counting question of Problem 297 and the minimum-term question of Problem 295.

Existing formalization

The Lean behind the site's (LEAN) suffix is the file src/latest/ErdosProblems/Erdos300.lean of Boris Alexeev's lean-proofs collection, posted 17 August 2026 and linked at its pinned commit on Liu and Sawhney's claim page. The file declares itself a formalization of a solution to Problem 300, names Liu and Sawhney as informal authors and Codex and GPT-5.6 Sol as formal authors, cites Theorem 1.3 of their paper, and proves erdos_300: the size of the largest unit-subsum-free subset of $\{1,\ldots,N\}$, divided by $N$, tends to $1-1/e$. The formal-conjectures statement file, added 20 September 2026, states erdos_300 in that form and the variant that the expected asymptotic $A(N)=(1+o(1))N$ of Erdős and Graham is false, both with sorry bodies, and tags that file's theorem as the formal proof of each. The community database lists the statement as formalized, and the site shows "Formalised statement? Yes". A vendored copy in Jayyhk/erdos-lean, posted 1 September 2026, proves the same erdos_300. The corpus has built and audited none of these, so the claim carries no formalized evidence. On 2026-09-17 the site showed "Formalised statement? No" and formal-conjectures had no file for the problem.

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_1980_old_new_problems_results_combinatorial_number_theory
• various_1999_some_pauls_favorite_problems
• croot_2003_coloring_conjecture_about_unit_fractions
• croot_2003_coloring_conjecture_about_unit_fractions / main_theorem
• liu_2024_further_questions_regarding_unit_fractions
• liu_2024_further_questions_regarding_unit_fractions / lemma_5_1
• liu_2024_further_questions_regarding_unit_fractions / lemma_6_2
• liu_2024_further_questions_regarding_unit_fractions / proposition_5_2
• liu_2024_further_questions_regarding_unit_fractions / theorem_1_3