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.