The largest sum-free subset of the lattice cube
Fix an integer $d \ge 1$ and let $f_d(N)$ be the size of the largest sum-free subset of $[N]^d$. The main part of the question is to determine $c_d = \limsup_{N \to \infty} N^{-d} f_d(N)$. It is easy that $c_1 = \frac12$; Elsholtz and Rackham showed $c_2 = \frac35$, and Lepsveridze and Sun determined $c_3$ and $c_4$. A slice $\{x : u \le x_1 + \dots + x_d < 2u\}$, with $u$ chosen to maximise its size, was believed to be optimal, as it is asymptotically for $d \le 4$.
Problem. For fixed $d$, how large can a sum-free subset of $[N]^d$ be?
Source: P. J. Cameron (unpublished note, 2002), and for d = 2 H. Aydinian in Research problems from the 19th British Combinatorial Conference (2005). Listed as Problem 6 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
- NT
- Keywords
- extremal combinatorics, lattice points, sum-free sets
References
@article{cameron2005bcc19,
author = {Cameron, Peter J.},
title = {Research problems from the 19th {B}ritish {C}ombinatorial {C}onference},
journal = {Discrete Mathematics},
volume = {293},
number = {1-3},
pages = {313--320},
year = {2005},
doi = {10.1016/j.disc.2005.02.006}
}
@article{elsholtz2017rackham,
author = {Elsholtz, Christian and Rackham, Laurence},
title = {Maximal sum-free sets of integer lattice grids},
journal = {Journal of the London Mathematical Society (2)},
volume = {95},
number = {2},
pages = {353--372},
year = {2017},
doi = {10.1112/jlms.12006}
}
@article{lepsveridze2026sun,
author = {Lepsveridze, Saba and Sun, Yihang},
title = {Size of the largest sum-free subset of $[n]^3$ and $[n]^4$},
journal = {International Mathematics Research Notices},
number = {9},
year = {2026},
doi = {10.1093/imrn/rnag081},
eprint = {2311.18289},
archivePrefix = {arXiv}
}
@misc{keevash2026lim,
author = {Keevash, Peter and Lim, Jeck},
title = {On the largest sum-free subset of the lattice cube},
year = {2026},
eprint = {2605.00816},
archivePrefix = {arXiv}
}
@misc{luo2026counting,
author = {Luo, Haoran},
title = {The number of sum-free subsets of lattice cubes},
year = {2026},
eprint = {2608.23544},
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 6; version of January 2026, most recent update December 2025},
year = {2026}
}
Resolution claims
We determine the limiting density of the largest sum-free subset of the lattice cube $\{1,2,\ldots,n\}^d$ for all $d$, thus resolving the natural conjecture that it is constructed by two appropriate hyperplane slices. Equivalently, we show that the largest measure of a sum-free subset of the hypercube $(0,1)^d\subset \mathbb{R}^d$ is attained by $\setcond{x\in (0,1)^d}{1\leq L(x)<2}$ for some linear map $L:\mathbb{R}^d\to \mathbb{R}$. …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-29 | Needs verification → Solution claimed | — | Recorded here: arXiv:2605.00816v2. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-01030, "The largest sum-free subset of the lattice cube", clopen.solutions.