Megaparsec-scale structure around the proto-cluster core SPT2349$-$56 at
  $z\,{=}\,4.3$
  We present an extensive ALMA spectroscopic follow-up programme of the
$z\,{=}\,4.3$ structure SPT2349$-$56, one of the most actively star-forming
proto-cluster cores known, to identify additional members using their [C{\sc
ii}] 158\,$\mu$m and \mbox{CO(4--3)} lines. In addition to robustly detecting
the 14 previously published galaxies in this structure, we identify a further
15 associated galaxies at $z\,{=}\,4.3$, resolving 55$\,{\pm}\,$5\,per cent of
the 870-$\mu$m flux density at 0.5\,arcsec resolution compared to 21\,arcsec
single-dish data. These galaxies are distributed into a central core containing
23 galaxies extending out to 300\,kpc in diameter, and a northern extension,
offset from the core by 400\,kpc, containing three galaxies. We discovered
three additional galaxies in a red {\it Herschel\/}-SPIRE source 1.5\,Mpc from
the main structure, suggesting the existence of many other sources at the same
redshift as SPT2349$-$56 that are not yet detected in the limited coverage of
our data. An analysis of the velocity distribution of the central galaxies
indicates that this region may be virialized with a mass of
(9$\pm$5)$\,{\times}\,$10$^{12}$\,M$_{\odot}$, while the two offset galaxy
groups are about 30 and 60\,per cent less massive and show significant velocity
offsets from the central group. We calculate the [C{\sc ii}] and far-infrared
number counts, and find evidence for a break in the [C{\sc ii}] luminosity
function. We estimate the average SFR density within the region of SPT2349$-$56
containing single-dish emission (a proper diametre of 720\,kpc), assuming
spherical symmetry, to be roughly
4$\,{\times}\,10^4$\,M$_{\odot}$\,yr$^{-1}$\,Mpc$^{-3}$; this may be an order
of magnitude greater than the most extreme examples seen in simulations.
