Wang 2026 667 806 upper bound erdos problem

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lemma_1: Wang's dilation lemma: for a finite set D of positive integers and a dilation factor m with mD inside [N], a subset of [N] with no solution of the unit-fraction equation of Problem 301 meets mD in at most the independence number of the unit-fraction hypergraph on D.

proposition_1: Wang's finite certificate: a table of the independence numbers of the unit-fraction hypergraph on each initial segment of the 29 nontrivial divisors of 720, ending at 11, which the manuscript certifies by an exact-arithmetic script in its appendix.

theorem_1: The manuscript's claimed upper bound for the extremal function of Problem 301, from a finite divisor certificate on the divisors of 720; an unrefereed manuscript that the site's discussion describes as AI-generated.

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Xinjun Wang, A 667/806 Upper Bound for Erdős Problem #301 on Unit-Fraction-Free Sets. Unpublished manuscript, dated May 27, 2026 on its title page, posted on ResearchGate (2026).

The copy read for this card is the nine-page ResearchGate manuscript (title page dated May 27, 2026; no arXiv identifier, no journal, no version history); Theorem 1 was read on the page image of p. 2 and the rest in the text layer. The manuscript says nothing about how it was written. The site's Problem 301 discussion thread (https://www.erdosproblems.com/forum/thread/301, read in the refresh of 2026-09-17T15:40Z) describes it otherwise: a comment by the account Woett, 12:26 on 04 Jul 2026, says the preprint "is AI-generated without any disclaimers to this effect", and a comment by the account KentaKitamura, 07:43 on the same day, dates it to May 2026 and records its bound. This card records that description as the site thread's statement, not as a finding of its own. No notice is printed in the manuscript (pp. 1--2 and 8--9 read; the title page's front matter is the title, the author, whose footnote gives an email address and an ORCID iD, and the date "May 27, 2026"); the ResearchGate posting (Source below) returned HTTP 403 when read, so no hosting page's terms were observed, and no publisher page exists; the term is unstated.

Let f(N) be the largest size of a subset A of [N] containing no distinct a, b_1, ..., b_k with 1/a = 1/b_1 + ... + 1/b_k. Theorem 1 claims f(N) <= (667/806 + o(1))N, about 0.8275N, improving the elementary bound (25/28 + o(1))N of Wouter van Doorn recorded on Bloom's Erdos Problems site; the trivial lower bound (1/2 + o(1))N comes from the interval (N/2, N]. The method is a finite-configuration dilation argument: Lemma 1 shows that for any finite set D of positive integers and any m with mD inside [N], |A ∩ mD| <= alpha(D), the independence number of the unit-fraction hypergraph H(D), and summing over disjoint dilates converts a finite certificate into a density bound. Proposition 1 tabulates the prefix independence numbers alpha(D_j) for the 29 nontrivial divisors of 720 = 2^4 3^2 5, with alpha(D) = 11, values the paper checks by exact integer arithmetic in the script of Appendix A; summing the forced omissions j - alpha(D_j) over the disjoint dilates gives a missing density of 139/806. For problem 301 this is the smallest upper-bound constant stated in a written manuscript read here; it is unrefereed, its finite certificate was not rerun here, and the site's discussion thread carries computational claims of smaller constants (the account rickyc reports 319/390 on 04 Jul 2026) without a written proof. The paper does not approach the value 1/2 that Erdős and Graham asked about.

Read status: claims checked for Theorem 1 (statement read clause by clause on the page image of p. 2), for the statement of Lemma 1 (p. 2) and for the statement and table of Proposition 1 (p. 3), both read on the page image; the exact-arithmetic script of Appendix A (pp. 6--9) was read as text and not rerun; no proof was checked, and nothing here is independently reviewed.

Source: https://www.researchgate.net/publication/405304408_A_667806_Upper_Bound_for_Erdos_Problem_301_on_Unit-Fraction-Free_Sets.

Bears on. #301: Theorem 1 claims the upper bound $f(N)\le(667/806+o(1))N$ for the problem's extremal function, in an unrefereed manuscript; Lemma 1 and Proposition 1 are steps of its argument. None of the three bears on whether $f(N)=(1/2+o(1))N$.

Results.

• Theorem 1: f(N) <= (667/806 + o(1))N (p. 2; an author's claim recorded as a qualified lead on #301).
• Lemma 1 (Dilation lemma): A meets each dilate mD inside {1, ..., N} in at most alpha(D) elements (p. 2).
• Proposition 1 (Finite certificate): prefix independence numbers of the 29 nontrivial divisors of 720, alpha(D) = 11 (p. 3).

Relation to E301

This source bears on Problem 301.

The paper uses Problem 301's f(N) and the same pairwise-distinct reciprocal equation (equation (1), p. 1). For an admissible A in [N], its claimed conclusion is |A| <= (667/806 + o(1))N (Theorem 1, p. 2), an author's claim in an unrefereed manuscript with no acceptance evidence, whose certificate was not rerun here and which the site thread describes as AI-generated; Problem 301 records it as a qualified lead, not as an established bound.

The argument encodes reciprocal identities in a finite hypergraph (equation (2), p. 2), and Lemma 1 (p. 2) supplies the step |A ∩ mD_j| <= alpha(D_j) for every dilate. For the 29 nontrivial divisors D of 720 (equation (3), p. 3), Proposition 1 (p. 3) tabulates the independence number of every initial segment D_j, ending at alpha(D) = 11; Appendix A (pp. 6--9) supplies witnesses and an exact branch-and-bound script using the integer identity 720/d = sum of 720/e over e in E. Lemmas 2--3 (p. 4) show that the dilates indexed by the valuation-restricted set M of equation (4) are disjoint and that M has density 120/403. Summing the prefix omissions in Section 4 (pp. 4--5) gives the exact weighted value 139/240 (equations (5)--(6), p. 5), hence the omission density 139/806. Section 5 (p. 5) discusses further configuration searches as a possibility, without an optimality claim. This provides a concrete finite-configuration template for another upper-bound argument. It does not establish Problem 301's proposed density 1/2: if the claim is accepted, then together with the upper-half construction (pp. 1--2) it leaves 1/2 <= liminf f(N)/N <= limsup f(N)/N <= 667/806.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.