(purple for 100$\mu$m, blue for 160$\mu$m, turquoise for 250$\mu$m, green for 350$\mu$m, orange for 500$\mu$m). The dashed, dot-dash, and three-dot-dash lines indicate the infrared luminosity of a galaxie of $3 \times 10^{10}$M$_\odot$ (our mass cut) at the center of the main sequence, a factor of 4 above it, and a factor of 10 above it, respectively.](LIRlim.eps)

Fig.\[SBcomp\] shows the luminosity limit corresponding to a detection at 5$\sigma$ at two wavelengths or more. This was computed using both the starburst and the main-sequence templates of the @Magdis2012b SED library. This library contains different templates for main-sequence and starburst galaxies. The main-sequence template evolves with redshift, but not the starburst one. The lines correspond to the highest luminosity limit found using these two templates for each wavelength, which is the most pessimistic case. We also computed the infrared luminosity associated with a galaxy of $3\times 10^{10}\,\rm M_\odot$, i.e., our mass limit, on the main sequence (dashed line), a factor of 4 above it (dot-dash line), and a factor of 10 above it (three-dot-dash line). All the M$_\star > 3\times 10^{10}\,\rm M_\odot$ strong starbursts (sSFR$>$10 sSFR$_{\rm MS}$) should thus be detected in two or more *Herschel* bands below z=4. There is only one 