tokens: the positions on the SED of the observational wavelengths for various instruments of interest here: 250, 350, and 500 μ m for the SPIRE instrument on board Herschel ; 450 and 850 μ m for the SCUBA-2 instrument at the James Clerk Maxwell telescope; ∼1100 μ m for AzTEC at the LMT; 1400 μ m for SPT ; and 850, 1400, and 3000 μ m for ALMA . We have also highlighted the 5 σ detection limits for such instruments (attained in the deepest large-scale surveys undertaken so far or in the near future): S 250 ≈ 35 mJy, S 350 ≈ 40 mJy, and S 500 ≈ 50 mJy for SPIRE; S 450 ≈ 8 mJy and S 850 ≈ 2 mJy for SCUBA-2 ; S 1100 ≈ 1 mJy for AzTEC ; S 1400 ≈ 20 mJy for SPT ; and S 850 ≈ 0.42, S 1400 ≈ 0.11, and S 3000 ≈ 0.02 mJy for ALMA (500 hr on 100 arcmin 2 ). The accurate determination of the spectroscopic redshift for a large sample of dusty galaxies is a major problem. In order to probe the bright end of the SFR function at z ≳ 3, a strategy could be to preselect high-redshift sources using flux/color criteria from surveys conducted with Herschel or SCUBA-2 , and then perform a more accurate photometric (or even spectroscopic) redshift determination with observations on source by, e.g., AzTEC and ALMA . Commonly used preselection criteria (e.g., Dowell et al. 2014 ; Asboth et al. 2016 ) for high-redshift sources based on Herschel photometry involve looking for “350-peakers,” which are defined as sources with S 350 / S 250 ≳ 1 and S 500 / S 350 ≲ 1, or “500-risers,” which are defined as sources with S 350 / S 250 ≳ 1 and S 500 / S 350 ≳ 1. Actually, the uncertainties in the flux measurements make the distinction between peakers and risers quite loose; moreover, at z ≳ 4, the channel at 250 μ m refers to rest-frame wavelengths λ ≲ 50 μ m, where details of the SED due to different dust properties and a possible contribution from an AGN component can be relevant. Thus, here, we mainly focus on the color S 500 / S 350 . In Figure 9 (bottom panel), we plot its evolution with redshift and compare it with the measurements from ALMA /ALESS by Swinbank et al. ( 2014 ), finding reasonable agreement within the large uncertainties. Sources with S 500 / S 350 ≳ 0.8 are high-redshift z ≳ 3 candidates. However, the rather high limiting fluxes of Herschel cannot probe the SFR function much above z ∼ 5, since even sources with SFR ψ ≳ 10 3 M ⊙ yr −1 are too faint to be detected. To go much beyond z ≳ 4, the preselection based on the color S 850 / S 450 from SCUBA-2 photometry is much more efficient. In Figure 9 (bottom panel), we plot its evolution with redshift and compare it with the measurements from SCUBA-2 by Koprowski et al. ( 2016 ), again finding reasonable agreement. It can be seen that the color condition S 850 / S 450 ≳ 0.6 can be exploited to preselect candidate galaxies at z ≳ 4. In Figure 10 , we show our predictions for the differential counts at 500 and 850 μ m for the red sources preselected according to the color criteria discussed above. At 500 μ m, the
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