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  1. Free, publicly-accessible full text available December 20, 2022
  2. We present images obtained with LABOCA on the APEX telescope of a sample of 22 galaxies selected via their red Herschel SPIRE 250-, 350- and $500\textrm{-}\mu\textrm{m}$ colors. We aim to see if these luminous, rare and distant galaxies are signposting dense regions in the early Universe. Our $870\textrm{-}\mu\textrm{m}$ survey covers an area of $\approx0.8\,\textrm{deg}^2$ down to an average r.m.s. of $3.9\,\textrm{mJy beam}^{-1}$, with our five deepest maps going $\approx2\times$ deeper still. We catalog 86 DSFGs around our 'signposts', detected above a significance of $3.5\sigma$. This implies a $100\pm30\%$ over-density of $S_{870}>8.5\,\textrm{mJy}$ DSFGs, excluding our signposts, when comparing our number countsmore »to those in 'blank fields'. Thus, we are $99.93\%$ confident that our signposts are pinpointing over-dense regions in the Universe, and $\approx95\%$ confident that these regions are over-dense by a factor of at least $\ge1.5\times$. Using template SEDs and SPIRE/LABOCA photometry we derive a median photometric redshift of $z=3.2\pm0.2$ for our signposts, with an interquartile range of $z=2.8\textrm{-}3.6$. We constrain the DSFGs likely responsible for this over-density to within $|\Delta z|\le0.65$ of their respective signposts. These 'associated' DSFGs are radially distributed within $1.6\pm0.5\,\textrm{Mpc}$ of their signposts, have median SFRs of $\approx(1.0\pm0.2)\times10^3\,M_{\odot}\,\textrm{yr}^{-1}$ (for a Salpeter stellar IMF) and median gas reservoirs of $\sim1.7\times10^{11}\,M_{\odot}$. These candidate proto-clusters have average total SFRs of at least $\approx (2.3\pm0.5)\times10^3\,M_{\odot}\,\textrm{yr}^{-1}$ and space densities of $\sim9\times10^{-7}\,\textrm{Mpc}^{-3}$, consistent with the idea that their constituents may evolve to become massive ETGs in the centers of the rich galaxy clusters we see today.« less