  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), galaxies
instead deviate downwards from the local $\Sigma$_[C II] - $\Sigma$_SFR
relation. This deviation might be caused by different interstellar medium (ISM)
properties in galaxies at early epochs. To test this hypothesis, we combined
the [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 deviation
from the $\Sigma$_[C II] - $\Sigma$_SFR relation in these sources. We present
the VLT/X-SHOOTER observations targeting the C III] ${\lambda}1909$ line
emission in three galaxies at 5.5 < z < 7. We include X-SHOOTER and VLT/MUSE
archival data of eight galaxies at 2 < z < 7, and eleven star-forming systems
at 6 < z < 7.5, with either C III] or [C II] detection reported in the
literature. We detected C III] ${\lambda}{\lambda}1907, 1909$ line emission in
HZ10 and we derived the intrinsic, integrated flux of the C III]
${\lambda}1909$ line. We constrained the ISM properties for our sample of
galaxies, $\kappa_s$ , $n$, and $Z$, by applying our physically motivated model
based on the MCMC algorithm. For the most part, high-z star-forming galaxies
show subsolar metallicities. The majority of the sources have $log(\kappa_s) >
1$, that is, they overshoot the Kennicutt-Schmidt (KS) relation by about one
order of magnitude, implying that the whole KS relation might be shifted
upwards at early times. Furthermore, all the high-z galaxies of our sample lie
below 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.
