Problem Statement
A set of integers $A$ is Ramsey $2$-complete if, whenever $A$ is $2$-coloured, all sufficiently large integers can be written as a monochromatic sum of elements of $A$.
Burr and Erdős [BuEr85] showed that there exists a constant $c>0$ such that it cannot be true that\[\lvert A\cap \{1,\ldots,N\}\rvert \leq c(\log N)^2\]for all large $N$ and that there exists a Ramsey $2$-complete $A$ such that for all large $N$\[\lvert A\cap \{1,\ldots,N\}\rvert < (2\log_2N)^3.\]Improve either of these bounds.
Burr and Erdős [BuEr85] showed that there exists a constant $c>0$ such that it cannot be true that\[\lvert A\cap \{1,\ldots,N\}\rvert \leq c(\log N)^2\]for all large $N$ and that there exists a Ramsey $2$-complete $A$ such that for all large $N$\[\lvert A\cap \{1,\ldots,N\}\rvert < (2\log_2N)^3.\]Improve either of these bounds.
Categories:
Number Theory Ramsey Theory
Progress
The stated bounds are due to Burr and Erdős [BuEr85]. Resolved by Conlon, Fox, and Pham [CFP21], who constructed a Ramsey $2$-complete $A$ such that\[\lvert A\cap \{1,\ldots,N\}\rvert \ll (\log N)^2\]for all large $N$.See also [55] and [843].
Source: erdosproblems.com/54 | Last verified: January 13, 2026