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∩[n]|≤Clog2n 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.