OP-00935 Open

Naïve corners in products of finite groups

Let $G$ be a finite group and $A \subset G \times G$ a set of density $\alpha$. Austin calls the configurations $\{(x, y), (gx, y), (x, gy)\}$ naïve corners. For the other type of corner, $\{(x, y), (xg, y), (x, gy)\}$ (BMZ corners, after Bergelson, McCutcheon and Zhang), Solymosi has answered the corresponding question.

Problem. Is it true that there are $\gg_\alpha |G|^3$ triples $(x, y, g) \in G^3$ such that $(x, y)$, $(gx, y)$ and $(x, gy)$ all lie in $A$, the implied constant depending only on $\alpha$ and not on $G$?

Source: T. Austin, Ajtai–Szemerédi theorems over quasirandom groups (2016), Question 2. Listed as Problem 18 in B. Green, 100 open problems (version of January 2026).

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

Classification

Primary: CO — Combinatorics
Secondary: GR
Keywords: corners, density theorems, finite groups, quasirandom groups
Literature checked: Oct. 26, 2026 by clemens-admin

References

@incollection{austin2016,
  author = {Austin, Tim},
  title = {Ajtai--{S}zemer{\'e}di theorems over quasirandom groups},
  booktitle = {Recent Trends in Combinatorics},
  series = {IMA Volumes in Mathematics and its Applications},
  volume = {159},
  pages = {453--484},
  publisher = {Springer},
  year = {2016},
  doi = {10.1007/978-3-319-24298-9_19},
  eprint = {1503.08746},
  archivePrefix = {arXiv}
}

@article{solymosi2013roth,
  author = {Solymosi, J{\'o}zsef},
  title = {Roth-type theorems in finite groups},
  journal = {European Journal of Combinatorics},
  volume = {34},
  number = {8},
  pages = {1454--1458},
  year = {2013},
  doi = {10.1016/j.ejc.2013.05.027}
}

@article{gowers2008quasirandom,
  author = {Gowers, W. T.},
  title = {Quasirandom groups},
  journal = {Combinatorics, Probability and Computing},
  volume = {17},
  number = {3},
  pages = {363--387},
  year = {2008},
  doi = {10.1017/S0963548307008826}
}

@misc{green100open,
  author = {Green, Ben},
  title = {100 open problems},
  howpublished = {\url{https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf}},
  note = {Problem 18; 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 It will not change if the problem is solved, reclassified or withdrawn.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-00935, "Naïve corners in products of finite groups", clopen.solutions.