Problem 312

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Statement. Does there exist some $c>0$ such that, for any $K>1$, whenever $A$ is a sufficiently large finite multiset of positive integers with $\sum_{n\in A}\frac{1}{n}>K$ there exists some $S\subseteq A$ such that

$$1-e^{-cK} < \sum_{n\in S}\frac{1}{n}\leq 1?$$

Formulation. The site's wording, accessed 2026-09-18 (page last edited 20 January 2026). $A$ is a finite multiset of positive integers, $S$ a submultiset, and the sums count multiplicity; write $R(A)=\sum_{n\in A}1/n$ and $\varepsilon(A)=1-\max{R(S):S\subseteq A,
R(S)\le1}$ for the deficit of the best subsum not exceeding $1$, so the question asks whether $R(A)>K$ forces $\varepsilon(A)<e^{-cK}$. The 1980 monograph puts the size condition on the number of terms ("for fixed $\alpha$ and $t$ sufficiently large, if $\sum_{k=1}^t1/s_k>\alpha$"); the site's "sufficiently large finite multiset" renders it; Korsky's theorem below has no size condition and instead requires $K\ge K_0$. Multiplicity matters: with distinct denominators the problem is different (a dense-set version is Problem 310).

Status. Open; the site labels it OPEN. Erdős and Graham state the bound with $c/K^2$ in place of $e^{-cK}$ (the monograph, without proof or reference); the best bound found is Korsky's $\varepsilon(A)\le\exp(-c\sqrt{K\log K})$ for $R(A)>K\ge K_0$, an arXiv preprint of July 2026 whose acknowledgment declares extensive assistance from GPT-5.5 Pro, with no journal record and, by its author's own statement, no journal submission planned; a construction (the site's thread; the preprint's display (1.4)) shows that no bound better than $\exp(-(1+o(1))K\log K)$ holds for all multisets. Korsky's bound settles no instance of the question, so it has no claim page. The conjectured exponential rate is open, and no proof, disproof or proof claim for it was found in the search whose scope the Current assessment records; this is a bounded negative finding.

Source. erdosproblems.com/312, accessed 2026-09-18: the problem page (labeled OPEN, with the site's standard note that no finite computation can settle it; source key [ErGr80] without a page; last edited 20 January 2026; "Formalised statement? Yes"), its eight-comment discussion thread and its empty proof-claim tab. The site thanks Mehtaab Sawhney. Cite as: T. F. Bloom, Erdős Problem #312, https://www.erdosproblems.com/312, accessed 2026-09-18.

References.

• [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. 40 (the site gives no page). Library home: erdos_1980_old_new_problems_results_combinatorial_number_theory.
• [Ko26] Korsky, S., A stretched-exponential bound for an Erdős--Graham unit-fraction problem. arXiv:2607.04157v1 (5 July 2026; title page dated 7 July 2026), 27 pages. Library home: korsky_2026_stretched_exponential_bound_erdos_graham_unit; result page for Theorem 1.1.

Formalization. Statement only. The file ErdosProblems/312.lean of formal-conjectures (main, 2026-09-18; the link is pinned to that commit) declares erdos_312, under category research open with proof sorry, as the equivalence of answer(sorry) with the following: there is a real $c>0$ such that for every real $K>1$ there is an $N_0$ with the property that for every $n\ge N_0$ and every function $a$ from Fin n to the natural numbers whose reciprocals (cast to the reals) sum to more than $K$, some Finset (Fin n) has reciprocal sum strictly above $1-\exp(-cK)$ and at most $1$. It encodes the multiset as a function on Fin n with $n\ge N_0$ (the monograph's condition on the number of terms) and does not require a i ≥ 1; no audit of the encoding against the site's statement is recorded. The community database (teorth/erdosproblems, data/problems.yaml, 2026-09-18) records status open (31 August 2025), a formalized statement (24 September 2025), formal status unformalized and no OEIS entry. No external Lean artifact is linked from the thread.

Current assessment

The question (site formulation of 2026-09-18). The statement above; OPEN, last edited 20 January 2026; source [ErGr80]. The whole commentary is one sentence, saying that Erdős and Graham knew the statement with $c/K^2$ in place of $e^{-cK}$. The thread (eight comments, none verified by the site): 18 August 2025, Kovač introduces $R(A)$ and $\epsilon(A)$ and the example below showing $\epsilon(A)\ge e^{-(1+o(1))R(A)\log R(A)}$ along a sequence of multisets, asking how optimal the exponential form is; 18 October 2025, Kovač reports that Gemini Deep Research and ChatGPT found no closely related literature (the linked transcript is not a source and is not cited on this page); 24 June 2026, Korsky announces a forthcoming preprint proving $\varepsilon(A)\le\exp(-c\sqrt{K\log K})$ and sketches the simpler $\exp(-c\sqrt K)$ argument (maximal subsum, compression to a stable multiset, a sparse activation lemma proved by Fourier analysis), with an acknowledgment that the Fourier details were performed by GPT-5.5 and a caution that they deserve close checking; 7 July 2026, the arXiv link; 9 July 2026, four comments in which Kovač questions the paper's terminology as machine-written and its crediting of sources, another commenter quotes the paper's AI acknowledgment and asks whether the author inspected the argument closely, and Korsky replies that this is his most AI-involved paper, that he reviewed it personally over several weeks and is confident in the result, that the section and idea names were produced by GPT-5.5, and that he has no plans to send it to a journal. The proof-claim tab is empty. The community database says open.

Origin. Printed p. 40 of the 1980 monograph: "Is it true that there is a $c>0$ such that for fixed $\alpha$ and $t$ sufficiently large, if $\sum_{k=1}^t1/s_k>\alpha$ then for some choice of $\varepsilon_k=0$ or $1$, $0\le1-\sum_{k=1}^t\varepsilon_k/s_k<e^{-c\alpha}$? We know only $c/\alpha^2$ as an upper bound at present." The $c/\alpha^2$ bound is stated without proof or reference, and no published proof of it was located; Korsky's introduction attributes it to the monograph ("They proved the polynomial estimate $\varepsilon(A)\ll K^{-2}$"). It is recorded on this page as the proposers' attested bound, not as a compiled result.

Best known bound (a preprint claim). Korsky's Theorem 1.1 (arXiv v1, p. 1): there are absolute constants $c>0$ and $K_0$ such that every finite multiset $A$ of positive integers with $R(A)>K\ge K_0$ satisfies $\varepsilon(A)\le\exp(-c\sqrt{K\log K})$, that is, some submultiset has reciprocal sum in $[1-\exp(-c\sqrt{K\log K}),1]$. The proof (Sections 2--4 and two appendices, 24 pages) removes a maximal subsum, compresses the remainder into a multiset $C$ with multiplicities $m_n<P^-(n)$ whose subsums lift to $A$ and avoid $(1-N^{-2},1)$ with $N=1/\varepsilon(A)$, and shows by a sparse-activation local-limit estimate that such a $C$ has reciprocal mass $R(C)\ll(\log N)^2/\log\log N$; since $R(C)>K-1$, this gives $\log N\gg\sqrt{K\log K}$. Read depth: claims checked for the theorem, the barrier display and the two compression lemmas, with their proofs; the analytic core in outline only. Acceptance: none. The paper is an unrefereed preprint; its acknowledgment declares that the author was assisted by GPT-5.5 Pro extensively during the development and writing of the manuscript, in particular in filling in technical details (the author claims the conceptual reductions and proof strategy and takes responsibility), the thread disputes its provenance and crediting as described above, and the author states he will not submit it to a journal. The site's page does not cite it. It is recorded on this page as the best bound found, with that provenance, and its correctness is neither confirmed nor doubted on this page.

The barrier. Let $A_z$ contain $p-1$ copies of $1/p$ for every prime $p\le z$ (Kovač, 18 August 2025; Korsky's display (1.4)). No submultiset has reciprocal sum exactly $1$, all subsums lie on a lattice of spacing $1/\prod_{p\le z}p=e^{-(1+o(1))z}$, and $R(A_z)\sim z/\log z$, so $\varepsilon(A_z)\ge\exp(-(1+o(1))R(A_z)\log R(A_z))$. Any bound valid for all multisets is therefore at best $\exp(-(1+o(1))K\log K)$; the conjectured $e^{-cK}$ is compatible with it, and the truth lies between $\exp(-(1+o(1))K\log K)$ (worst case at least this large) and $\exp(-c\sqrt{K\log K})$. The argument is elementary and is recorded as the sources give it, not independently rechecked.

Formal statements. The formal-conjectures statement is summarized under Formalization; nothing beyond it exists.

Search scope. The problem, discussion and proof-claim pages; the community database record; the formal-conjectures file at the pinned commit; the arXiv listing for 2607.04157 (v1 only, no journal reference) and a Crossref bibliographic query for the title (no record); the Semantic Scholar record (rate-limited, no data); arXiv API searches for abstracts naming unit fractions and Graham (seven records, none on this deficit question), Egyptian fractions with subset sums or multisets (one unrelated record) and "Erdos problem" with the problem number (none); the monograph's p. 40; one general web search (Korsky's arXiv pages, a Wikipedia article on a different Erdős--Graham problem, a Discrete Analysis paper on Egyptian fractions of Problem 297's kind; nothing else). Not searched: MathSciNet, zbMATH, Google Scholar full text, X. Nothing found proves the exponential bound or refutes it; this is a bounded negative finding.

Remaining gaps. (1) Korsky's analytic core is recorded in outline only, and the paper is unrefereed and unreviewed; it is the only source for the stretched exponential. (2) The monograph's $c/K^2$ has no located proof. (3) Whether the true worst-case rate is exponential in $K$ or of order $K\log K$ in the exponent is open. (4) The formal statement's encoding was not audited.

Progress and known results

• Erdős and Graham (1980, printed p. 40): the question; the bound $c/K^2$ stated without proof.
• Korsky (2026, preprint): Theorem 1.1, $\varepsilon(A)\le\exp(-c\sqrt{K\log K})$ for $R(A)>K\ge K_0$; display (1.4), the barrier $\exp(-(1+o(1))K\log K)$ from the prime multisets.
• Related: the dense-subset version with bounded denominators, Problem 310; the near-one subsums of $\{1,\ldots,N\}$ of Problem 311; the exact representations of $1$ from short intervals of Problem 286.

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
• korsky_2026_stretched_exponential_bound_erdos_graham_unit
• korsky_2026_stretched_exponential_bound_erdos_graham_unit / theorem_1_1