Large subsets of [N] with no solution of x + 3y = 2z + 2w in distinct integers
Let $f(N)$ be the size of the largest subset of $[N]$ containing no solution of $x + 3y = 2z + 2w$ in distinct integers $x, y, z, w$. The bounds known to Green are $N^{1/2} \ll f(N) \ll N e^{-c(\log N)^{1/7}}$, the lower bound due to Ruzsa and the upper bound to Schoen and Sisask.
Problem. Determine $f(N)$.
Source: I. Z. Ruzsa, Solving a linear equation in a set of integers I, Acta Arith. 65 (1993), Section 9. Listed as Problem 16 in B. Green, 100 open problems (version of January 2026).
Imported from people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf.
Classification
- Primary
- NT — Number Theory
- Secondary
- CO
- Keywords
- linear equations in sets, Ruzsa's problem, solution-free sets
References
@article{ruzsa1993linear,
author = {Ruzsa, Imre Z.},
title = {Solving a linear equation in a set of integers {I}},
journal = {Acta Arithmetica},
volume = {65},
number = {3},
pages = {259--282},
year = {1993},
doi = {10.4064/aa-65-3-259-282}
}
@article{schoen2016sisask,
author = {Schoen, Tomasz and Sisask, Olof},
title = {Roth's theorem for four variables and additive structures in sums of sparse sets},
journal = {Forum of Mathematics, Sigma},
volume = {4},
pages = {e5},
year = {2016},
doi = {10.1017/fms.2016.2}
}
@misc{hametner2026solutionfree,
author = {Hametner, Paul and Solymosi, Jozsef},
title = {Large solution-free sets via combinatorial degenerations},
year = {2026},
eprint = {2609.13972},
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 16; version of January 2026, most recent update December 2025},
year = {2026}
}
Resolution claims
Consider the linear form $L(x,y,z,w) = 3x+y-2z-2w$. For a positive integer $N$, denote by $r_L(N)$ the largest size of a subset of $\{1,2,\dots,N\}$ that avoids nontrivial solutions to $L = 0$. We show that $r_L(N) = Ω(N^{0.5608687})$, improving the lower bound for Problem 16 in Green's list of open problems. Our proof uses the method of combinatorial degenerations to turn …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-29 | Open → Partially resolved | The registry's curators | Large solution-free sets via combinatorial degenerations v1 recorded from arXiv:2609.13972v1. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-01032, "Large subsets of [N] with no solution of x + 3y = 2z + 2w in distinct integers", clopen.solutions.