  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 $z \approx
3$.
