A negative answer to Erdős Problem #786
Abstract
Call a set A of positive integers admissible if, whenever a1⋯ ar=b1⋯ bs with a1,…,ar distinct elements of A and b1,…,bs distinct elements of A, necessarily r=s. Erdős asked whether admissible sets can have density 1−ε for every ε>0, and whether {1,…,N} always contains an admissible subset of size $(1-o(1))N$. For the variant in which repetitions are allowed both questions were answered negatively by Erdős, Ruzsa and Sárközy and by Granville and Soundararajan; for products of distinct elements, the first question was answered only recently (with density bound 7/8), and the second has remained open. We show that every admissible A⊆{1,…,N} satisfies ∑a∈ A1/a≤1/2log N+(loglog N+2)2, and that there is an absolute constant η>0 such that every admissible A⊆{1,…,N} has |A|<(1−η)N for all large N. Both questions therefore have negative answers. The proofs are elementary; the second rests on a coupling that replaces the largest divisor of an integer composed of small primes, which avoids the divisor-function losses inherent in counting quotients along a multiplication table. Both negative answers are formally verified in Lean 4 against the statements of the Formal Conjectures project.
Related
- A negative answer to the Erdős-Sárkőzy question
- Infinite Generalized Sidon Sets with a Negative Answer to a Problem of Szemerédi
- Quadratic bounds for intersecting hypergraphs: An affirmative answer to a question of Duke and Erdős
- Self-dual sets up to a translation: a negative answer to Milman's question
- A negative answer to a question on tilting objects and two-term complexes
BibTeX
@misc{li2026negative,
title = {{A negative answer to Erdős Problem #786}},
author = {Shisheng Li},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.37471},
eprint = {2609.37471},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.37471},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A negative answer to Erdős Problem #786
AU - Shisheng Li
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.37471
UR - https://arxiv.org/abs/2609.37471
AB - Call a set A of positive integers admissible if, whenever a_1⋯ a_r=b_1⋯ b_s with a_1,…,a_r distinct elements of A and b_1,…,b_s distinct elements of A, necessarily r=s. Erdős asked whether admissible sets can have density 1−ε for every ε>0, and whether {1,…,N} always contains an admissible subset of size $(1-o(1))N$. For the variant in which repetitions are allowed both questions were answered negatively by Erdős, Ruzsa and Sárközy and by Granville and Soundararajan; for products of distinct elements, the first question was answered only recently (with density bound 7/8), and the second has remained open. We show that every admissible A⊆{1,…,N} satisfies ∑_a∈ A1/a≤1/2log N+(loglog N+2)^2, and that there is an absolute constant η>0 such that every admissible A⊆{1,…,N} has |A|<(1−η)N for all large N. Both questions therefore have negative answers. The proofs are elementary; the second rests on a coupling that replaces the largest divisor of an integer composed of small primes, which avoids the divisor-function losses inherent in counting quotients along a multiplication table. Both negative answers are formally verified in Lean 4 against the statements of the Formal Conjectures project.
ER -