Ramsey complete sequences

A sequence of positive integers $A$ is said to be entirely Ramsey complete if, for any two-colouring of $A$, every positive integer can be written as the sum of distinct elements of $A$ of the same colour. We show that there exists a constant $C$ and an entirely Ramsey complete sequence $A$ such that $|A \cap [n]| \leq C \log^2n$ for all $n$. This is best possible up to the constant and solves a problem of Burr and Erd\H{o}s. We also discuss several related problems stated by the same authors. Joint work with Jacob Fox.Non UBCUnreviewedAuthor affiliation: California Institute of TechnologyFacult