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A tight relation between [C II] line luminosity and the star formation rate(SFR) has been observed for local galaxies. At high redshift (z > 5), galaxiesinstead deviate downwards from the local $\Sigma$_[C II] - $\Sigma$_SFRrelation. This deviation might be caused by different interstellar medium (ISM)properties in galaxies at early epochs. To test this hypothesis, we combinedthe [C II] and SFR data with C III] line observations and our physical models.We additionally investigated how ISM properties, such as burstiness,$\kappa_s$, total gas density, $n$, and metallicity, $Z$, affect the deviationfrom the $\Sigma$_[C II] - $\Sigma$_SFR relation in these sources. We presentthe VLT/X-SHOOTER observations targeting the C III] ${\lambda}1909$ lineemission in three galaxies at 5.5 < z < 7. We include X-SHOOTER and VLT/MUSEarchival data of eight galaxies at 2 < z < 7, and eleven star-forming systemsat 6 < z < 7.5, with either C III] or [C II] detection reported in theliterature. We detected C III] ${\lambda}{\lambda}1907, 1909$ line emission inHZ10 and we derived the intrinsic, integrated flux of the C III]${\lambda}1909$ line. We constrained the ISM properties for our sample ofgalaxies, $\kappa_s$ , $n$, and $Z$, by applying our physically motivated modelbased on the MCMC algorithm. For the most part, high-z star-forming galaxiesshow subsolar metallicities. The majority of the sources have $log(\kappa_s) >1$, that is, they overshoot the Kennicutt-Schmidt (KS) relation by about oneorder of magnitude, implying that the whole KS relation might be shiftedupwards at early times. Furthermore, all the high-z galaxies of our sample liebelow the $\Sigma$_[C II] - $\Sigma$_SFR local relation. The total gas density,$n$, shows the strongest correlation with the deviation from the local$\Sigma$_[C II] - $\Sigma$_SFR relation.

The interstellar medium of high-redshift galaxies: Gathering clues from C III] and [C II] lines