OP-00981 Open

Power-saving bounds for four-term progressions in F_5^n

Let $r_4(\mathbb{F}_5^n)$ be the largest size of a subset of $\mathbb{F}_5^n$ containing no four-term arithmetic progression with nonzero common difference, and let $N = 5^n$. A set with no three-term progression has no four-term progression, so a positive answer would include the power-saving bound of Ellenberg and Gijswijt, based on the method of Croot, Lev and Pach, for three-term progressions in $\mathbb{F}_5^n$; extending that method to four-term progressions has not succeeded so far.

Problem. Is $r_4(\mathbb{F}_5^n) \ll N^{1-c}$ for some constant $c > 0$?

Source: B. Green, 100 open problems (version of January 2026), Problem 9(iii).

Imported from people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf.

Classification

Primary: CO — Combinatorics
Secondary: NT
Keywords: arithmetic progressions, finite field model, polynomial method, Szemerédi's theorem
Literature checked: Oct. 26, 2026 by clemens-admin

References

@article{green2009tao,
  author = {Green, Ben and Tao, Terence},
  title = {New bounds for {S}zemer{\'e}di's theorem, {I}: progressions of length 4 in finite field geometries},
  journal = {Proceedings of the London Mathematical Society (3)},
  volume = {98},
  number = {2},
  pages = {365--392},
  year = {2009},
  doi = {10.1112/plms/pdn030}
}

@misc{green2012tao,
  author = {Green, Ben and Tao, Terence},
  title = {New bounds for {S}zemer{\'e}di's theorem, {Ia}: progressions of length 4 in finite field geometries revisited},
  year = {2012},
  eprint = {1205.1330},
  archivePrefix = {arXiv}
}

@incollection{peluse2024survey,
  author = {Peluse, Sarah},
  title = {Finite field models in arithmetic combinatorics -- twenty years on},
  booktitle = {Surveys in Combinatorics 2024},
  pages = {159--200},
  publisher = {Cambridge University Press},
  year = {2024},
  doi = {10.1017/9781009490559.007}
}

@article{ellenberg2017gijswijt,
  author = {Ellenberg, Jordan S. and Gijswijt, Dion},
  title = {On large subsets of $\mathbb{F}_q^n$ with no three-term arithmetic progression},
  journal = {Annals of Mathematics (2)},
  volume = {185},
  number = {1},
  pages = {339--343},
  year = {2017},
  doi = {10.4007/annals.2017.185.1.8}
}

@misc{karingula2026slice,
  author = {Karingula, Sankeerth Rao and Lovett, Shachar},
  title = {Limitations of the slice rank method in additive combinatorics},
  year = {2026},
  eprint = {2609.27044},
  archivePrefix = {arXiv}
}

@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}
}

Citing this problem

This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.

OP-00981, "Power-saving bounds for four-term progressions in F_5^n", clopen.solutions.