String correlators on ${\text{AdS}}_3$: Three-point functions

We revisit the computation of string worldsheet correlators on Euclidean ${\text{AdS}}_3$ with pure NS-NS background. We compute correlation functions with insertions of spectrally flowed operators. We explicitly solve all the known constraints of the model and for the first time conjecture a closed formula for three-point functions with arbitrary amount of spectral flow. We explain the relation of our results with previous computations in the literature and derive the fusion rules of the model. This paper is the first in a series with several installments.