Non-spherical sets versus lines in Euclidean Ramsey theory

We show that for every non-spherical set X in Eᵈ, there exists a natural number m and a red/blue-colouring of Eⁿ for every n such that there is no red copy of X and no blue progression of length m with each consecutive point at distance 1. This verifies a conjecture of Wu and the first author.