  Mapping of the large-scale structure through cosmic time has numerous
applications in the studies of cosmology and galaxy evolution. At $z > 2$, the
structure can be traced by the neutral intergalactic medium (IGM) by way of
observing the Ly$\alpha$, forest towards densely-sampled lines-of-sight of
bright background sources, such as quasars and star forming galaxies. We
investigate the scientific potential of MOSAIC, a planned multi-object
spectrograph on the European Extremely Large Telescope (ELT), for the 3D
mapping of the IGM at $z \gtrsim 3$. We simulate a survey of $3 \lesssim z
\lesssim 4$ galaxies down to a limiting magnitude of $m_{r}\sim 25.5$ mag in an
area of 1 degree$^2$ in the sky. Galaxies and their spectra (including the
line-of-sight Ly$\alpha$ absorption) are taken from the lightcone extracted
from the Horizon-AGN cosmological hydrodynamical simulation. The quality of the
reconstruction of the original density field is studied for different spectral
resolutions and signal-to-noise ratios of the spectra. We demonstrate that the
minimum $S/N$ (per resolution element) of the faintest galaxies that such
survey has to reach is $S/N = 4$. We show that a survey with such sensitivity
enables a robust extraction of cosmic filaments and the detection of the
theoretically-predicted galaxy stellar mass and star-formation rate gradients
towards filaments. By simulating the realistic performance of MOSAIC we obtain
$S/N(T_{\rm obs}, R, m_{r})$ scaling relations. We estimate that $\lesssim
35~(65)$ nights of observation time are required to carry out the survey with
the instrument's high multiplex mode and with the spectral resolution of
$R=1000~(2000)$. A survey with a MOSAIC-concept instrument on the ELT is found
to enable the mapping of the IGM at $z > 3$ on Mpc scales, and as such will be
complementary to and competitive with other planned IGM tomography surveys.
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