Problem Statement
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?
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?
Categories:
Number Theory
Progress
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].
Source: erdosproblems.com/843 | Last verified: January 16, 2026