A polylogarithmic bound for sets without five-term progressions
Let $r_5(N)$ be the largest size of a subset of $\{1, \dots, N\}$ containing no nontrivial five-term arithmetic progression. Gowers proved $r_5(N) \ll N (\log \log N)^{-c}$ for some $c > 0$, and Leng, Sah and Sawhney improved this to $r_5(N) \ll N \exp(-(\log \log N)^{c})$. For three-term progressions, polylogarithmic savings and more are known (Kelley and Meka).
Problem. Is there a constant $c > 0$ with $r_5(N) \ll N (\log N)^{-c}$?
Source: B. Green, 100 open problems (version of January 2026), Problem 9(ii).
Imported from people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf.
Classification
- Primary
- CO — Combinatorics
- Secondary
- NT
- Keywords
- additive combinatorics, arithmetic progressions, Szemerédi's theorem
- Literature checked
- Nov. 3, 2026 by clemens-admin
References
@article{gowers2001szemeredi,
author = {Gowers, W. T.},
title = {A new proof of {S}zemer{\'e}di's theorem},
journal = {Geometric and Functional Analysis},
volume = {11},
number = {3},
pages = {465--588},
year = {2001},
doi = {10.1007/s00039-001-0332-9}
}
@misc{lengsahsawhney2024szemeredi,
author = {Leng, James and Sah, Ashwin and Sawhney, Mehtaab},
title = {Improved bounds for {S}zemer{\'e}di's theorem},
year = {2024},
eprint = {2402.17995},
archivePrefix = {arXiv}
}
@misc{kelleymeka2023,
author = {Kelley, Zander and Meka, Raghu},
title = {Strong bounds for 3-progressions},
year = {2023},
eprint = {2302.05537},
archivePrefix = {arXiv}
}
@misc{openai2026progressions,
author = {{OpenAI}},
title = {Quasipolynomial Bounds for Arithmetic Progressions},
howpublished = {\url{https://github.com/openai/math/blob/4eaee87548250b2682affda7d05d947c05554bae/preprints/Quasipolynomial-Bounds-for-Arithmetic-Progressions-September-23-2026/paper.pdf}},
note = {AI-generated, unrefereed; dated 23 September 2026},
year = {2026}
}
@misc{green100open,
author = {Green, Ben},
title = {100 open problems},
howpublished = {\url{https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf}},
note = {Problem 9; version of January 2026, most recent update December 2025},
year = {2026}
}
Resolution claims
We prove Erdős's conjecture that every set of positive integers with divergent reciprocal sum contains arithmetic progressions of every finite length. More quantitatively, for every fixed $k\ge3$, we show \[ r_k(N) C_kN(-c_k( N)^{_k}) \] with $C_k,c_k,\varepsilon_k>0$, where $r_k(N)$ is the largest size of a subset of $\{1,\ldots,N\}$ with no nonconstant $k$-term arithmetic progression.
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-11-08 | Needs verification → Solution claimed | — | Recorded here: math @ 4eaee87, build/main.tex. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-01132, "A polylogarithmic bound for sets without five-term progressions", clopen.solutions.