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 instruments 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. [abridged]
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