PROVED
This has been solved in the affirmative.
Are the squares Ramsey $2$-complete?
That is, is it true that, in any 2-colouring of the square numbers, every sufficiently large $n\in \mathbb{N}$ can be written as a monochromatic sum of distinct squares?
A problem of Burr and Erdős. A similar question can be asked for the set of $k$th powers for any $k\geq 3$.
In
[Er95] Erdős reported that Burr had proved that the set of $k$th powers is Ramsey $r$ complete for all $r,k\geq 2$, but this result was never published. A stronger version was proved by Conlon, Fox, and Pham
[CFP21], who proved that in fact the set of $k$th powers contains a sparse Ramsey $r$-complete subsequence, again for every $r,k\geq 2$.
See also
[54] and
[55].
View the LaTeX source
Additional thanks to: Sarosh Adenwalla
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 #843, https://www.erdosproblems.com/843, accessed 2026-01-16
