No‐(k+1)‐in‐line problem for large constant k
Grebennikov A, Kwan MA. 2026. No‐(k+1)‐in‐line problem for large constant k. Bulletin of the London Mathematical Society. 58(10), e70510.
Download
Journal Article
| Published
| English
Scopus indexed
Corresponding author has ISTA affiliation
Department
Abstract
How many points can be placed in an 𝑛 × 𝑛 grid sothat every (affine) line contains at most 𝑘 points? Weprove that for 𝑛 ⩾ 𝑘 ⩾ 1037 the maximum number ofpoints is exactly 𝑘𝑛. Our proof builds on the recent workof Kovács, Nagy and Szabó (who proved an analogousresult when 𝑘 is at least about √𝑛 log 𝑛), incorporat-ing ideas of Jain and Pham. Using the same approach,we also obtain new bounds for higher-dimensionalextensions of this problem
Publishing Year
Date Published
2026-10-01
Journal Title
Bulletin of the London Mathematical Society
Publisher
Wiley
Acknowledgement
We would like to thank Huy Pham for insightful discussions about the proof approach, and wewould like to thank the anonymous referees for their careful reading and helpful comments. Bothauthors are supported by ERC Starting Grant “RANDSTRUCT” No. 101076777.Open Access funding provided by Institute of Science and Technology Austria.
Volume
58
Issue
10
Article Number
e70510
ISSN
eISSN
IST-REx-ID
Cite this
Grebennikov A, Kwan MA. No‐(k+1)‐in‐line problem for large constant k. Bulletin of the London Mathematical Society. 2026;58(10). doi:10.1112/blms.70510
Grebennikov, A., & Kwan, M. A. (2026). No‐(k+1)‐in‐line problem for large constant k. Bulletin of the London Mathematical Society. Wiley. https://doi.org/10.1112/blms.70510
Grebennikov, Aleksandr, and Matthew Alan Kwan. “No‐(K+1)‐in‐line Problem for Large Constant K.” Bulletin of the London Mathematical Society. Wiley, 2026. https://doi.org/10.1112/blms.70510.
A. Grebennikov and M. A. Kwan, “No‐(k+1)‐in‐line problem for large constant k,” Bulletin of the London Mathematical Society, vol. 58, no. 10. Wiley, 2026.
Grebennikov A, Kwan MA. 2026. No‐(k+1)‐in‐line problem for large constant k. Bulletin of the London Mathematical Society. 58(10), e70510.
Grebennikov, Aleksandr, and Matthew Alan Kwan. “No‐(K+1)‐in‐line Problem for Large Constant K.” Bulletin of the London Mathematical Society, vol. 58, no. 10, e70510, Wiley, 2026, doi:10.1112/blms.70510.
All files available under the following license(s):
Creative Commons Attribution 4.0 International Public License (CC-BY 4.0):
Main File(s)
File Name
Access Level
Open Access
Date Uploaded
2026-10-14
MD5 Checksum
cce830e3fc66174803b91103f5b88a29
Google Scholar