We present 1.3- and/or 3-mm continuum images and 3-mm spectral scans, obtained using NOEMA and ALMA, of 21 distant, dusty, star-forming galaxies (DSFGs). Our sample is a subset of the galaxies selected by Ivison et al. (2016) on the basis of their extremely red far-infrared (far-IR) colours and low {it Herschel} flux densities; most are thus expected to be unlensed, extraordinarily luminous starbursts at $z gtrsim 4$, modulo the considerable cross-section to gravitational lensing implied by their redshift. We observed 17 of these galaxies with NOEMA and four with ALMA, scanning through the 3-mm atmospheric window. We have obtained secure redshifts for seven galaxies via detection of multiple CO lines, one of them a lensed system at $z=6.027$ (two others are also found to be lensed); a single emission line was detected in another four galaxies, one of which has been shown elsewhere to lie at $z=4.002$. Where we find no spectroscopic redshifts, the galaxies are generally less luminous by 0.3-0.4 dex, which goes some way to explaining our failure to detect line emission. We show that this sample contains amongst the most luminous known star-forming galaxies. Due to their extreme star-formation activity, these galaxies will consume their molecular gas in $lesssim 100$ Myr, despite their high molecular gas masses, and are therefore plausible progenitors of the massive, `red-and-dead elliptical galaxies at $z approx 3$.
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