DISPROVED
This has been solved in the negative.
Let $A\subset \mathbb{N}$ be an infinite set for which there exists some $\epsilon>0$ such that in any subset of $A$ of size $n$ there is a subset of size at least $\epsilon n$ which contains no three-term arithmetic progression.
Is it true that $A$ is the union of a finite number of sets which contain no three-term arithmetic progression?
A problem of Erdős, Nešetřil, and Rödl.
A negative answer was given by Reiher, Rödl, and Sales
[RRS24], who proved that, for any $0<\mu<1/2$, there exists $A\subseteq \mathbb{N}$ such that every finite colouring of $A$ contains a three-term arithmetic progression, and yet every subset of $A$ of size $n$ contains a subset of size $\geq \mu n$ without a three-term arithmetic progression.
See also
[774] and
[846].
View the LaTeX source
This page was last edited 27 January 2026. View history
Additional thanks to: Neel Somani and Andrew Xu
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 #847, https://www.erdosproblems.com/847, accessed 2026-04-11
