OPEN
This is open, and cannot be resolved with a finite computation.
Let $g(n)$ be maximal such that given any set $A\subset \mathbb{R}$ with $\lvert A\rvert=n$ there exists some $B\subseteq A$ of size $\lvert B\rvert\geq g(n)$ such that $b_1+b_2\not\in A$ for all $b_1\neq b_2\in B$.
Estimate $g(n)$.
Disclaimer: The open status of this problem reflects the current belief of the owner of this website. There may be literature on this problem that I am unaware of, which may partially or completely solve the stated problem. Please do your own literature search before expending significant effort on solving this problem. If you find any relevant literature not mentioned here, please add this in a comment.
This function was considered by Erdős and Moser. Choi observed that, without loss of generality, one can assume that $A\subset \mathbb{Z}$.
Klarner proved $g(n) \gg \log n$ (indeed, a greedy construction suffices). Choi
[Ch71] proved $g(n) \ll n^{2/5+o(1)}$. The current best bounds known are\[(\log n)^{1+c} \ll g(n) \ll \exp(\sqrt{\log n})\]for some constant $c>0$, the lower bound due to Sanders
[Sa21] and the upper bound due to Ruzsa
[Ru05]. Beker
[Be25] has proved\[(\log n)^{1+\tfrac{1}{68}+o(1)} \ll g(n).\]
View the LaTeX source
This page was last edited 23 January 2026. View history
|
Likes this problem
|
None
|
|
Interested in collaborating
|
None
|
|
Currently working on this problem
|
None
|
|
This problem looks difficult
|
Aron
|
|
This problem looks tractable
|
None
|
|
The results on this problem could be formalisable
|
None
|
|
I am working on formalising the results on this problem
|
None
|
Additional thanks to: Wouter van Doorn
When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #787, https://www.erdosproblems.com/787, accessed 2026-04-11
