Divisible point sets as unions of parallel lines
Let $p$ be a prime. A cylinder in $\mathbb{F}_p^3$ is a union of $p$ distinct parallel affine lines; it has $p^2$ points and meets every affine plane in a multiple of $p$ points. The statement below holds for $p \le 7$ (De Beule, Demeyer, Mattheus and Sziklai; Kurz and Mattheus, who corrected the earlier proof for $p = 5$ and gave the first for $p = 7$), and it holds for every set as below that contains a whole affine line (Blokhuis; see Kiss, Markó, Nagy and Somlai). Kiss, Markó, Nagy and Somlai proved that the indicator function of every such set is an integer linear combination of indicator functions of cylinders.
Problem. Let $S \subseteq \mathbb{F}_p^3$ contain exactly $p^2$ points, and suppose that $|S \cap H| \equiv 0 \pmod p$ for every affine plane $H \subseteq \mathbb{F}_p^3$. Must $S$ be the union of $p$ distinct parallel affine lines, that is, must there be $v \in \mathbb{F}_p^3 \setminus \{0\}$ and $a_1, \dots, a_p \in \mathbb{F}_p^3$ with $$S = \bigsqcup_{j=1}^{p} \bigl(a_j + \mathbb{F}_p v\bigr)?$$
Source: Ball (2008), the strong cylinder conjecture; see Kurz and Mattheus (2022), Conjecture 1, and Kiss, Markó, Nagy and Somlai (2026), Conjecture 1.6.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- CO — Combinatorics
- Secondary
- IT
- Keywords
- AIM Open Applied Problems, direction problems, divisible codes, finite geometry
References
@article{ball2008,
author = {Ball, Simeon},
title = {On the graph of a function in many variables over a finite field},
journal = {Des. Codes Cryptogr.},
volume = {47},
number = {1-3},
year = {2008},
pages = {159--164},
doi = {10.1007/s10623-007-9108-z},
}
@article{kurz2022,
author = {Kurz, Sascha and Mattheus, Sam},
title = {A generalization of the cylinder conjecture for divisible codes},
journal = {IEEE Trans. Inform. Theory},
volume = {68},
number = {4},
year = {2022},
pages = {2281--2289},
doi = {10.1109/TIT.2021.3134201},
eprint = {2011.02923},
archivePrefix = {arXiv},
}
@article{debeule2019,
author = {De Beule, Jan and Demeyer, Jeroen and Mattheus, Sam and Sziklai, P{\'e}ter},
title = {On the cylinder conjecture},
journal = {Des. Codes Cryptogr.},
volume = {87},
number = {4},
year = {2019},
pages = {879--893},
doi = {10.1007/s10623-018-0571-5},
}
@misc{kiss2026,
author = {Kiss, Gergely and Mark{\'o}, {\'A}d{\'a}m and Nagy, Zolt{\'a}n L{\'o}r{\'a}nt and Somlai, G{\'a}bor},
title = {Cylinder type and $p$-divisible sets in $\mathbb{F}_p^3$},
year = {2026},
eprint = {2601.09910},
archivePrefix = {arXiv},
}
@misc{aim-strong-cylinder,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Divisible point sets as unions of parallel lines}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/604-strong-cylinder.md},
note = {Entry 604 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00391, "Divisible point sets as unions of parallel lines", clopen.solutions.