Rado's boundedness conjecture
Let $a_1, \dots, a_k$ be integers that do not satisfy Rado's condition: no nonempty subset of them sums to $0$. By Rado's theorem the equation $a_1 x_1 + \dots + a_k x_k = 0$ is then not partition regular. Let $c(a_1, \dots, a_k)$ be the least number of colours in a colouring of $\mathbb{N}$ with no monochromatic solution of this equation. Fox and Kleitman proved $c(a_1, a_2, a_3) \le 24$, which settles the case $k = 3$; the cases $k \ge 4$ are open.
Problem. Is $c(a_1, \dots, a_k)$ bounded in terms of $k$ only?
Source: R. Rado, Studien zur Kombinatorik, Math. Z. 36 (1933). Listed as Problem 21 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
- Keywords
- colourings, linear equations, partition regularity, Rado's theorem
- Literature checked
- Oct. 26, 2026 by clemens-admin
References
@article{rado1933,
author = {Rado, Richard},
title = {Studien zur {K}ombinatorik},
journal = {Mathematische Zeitschrift},
volume = {36},
number = {1},
pages = {424--470},
year = {1933},
doi = {10.1007/BF01188632}
}
@article{fox2006kleitman,
author = {Fox, Jacob and Kleitman, Daniel J.},
title = {On {R}ado's boundedness conjecture},
journal = {Journal of Combinatorial Theory, Series A},
volume = {113},
number = {1},
pages = {84--100},
year = {2006},
doi = {10.1016/j.jcta.2005.07.004}
}
@misc{ellis2023leader,
editor = {Ellis, David and Johnson, Robert},
title = {A collection of open problems in celebration of {I}mre {L}eader's 60th birthday},
year = {2023},
eprint = {2310.18163},
archivePrefix = {arXiv},
note = {Mili{\'c}evi{\'c}, Conjecture 11.1}
}
@misc{green100open,
author = {Green, Ben},
title = {100 open problems},
howpublished = {\url{https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf}},
note = {Problem 21; 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 It will not change if the problem is solved, reclassified or withdrawn.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-00936, "Rado's boundedness conjecture", clopen.solutions.