Extremely metal-poor stars provide us with indirect information on the first generations of massive stars. The TOPoS survey has been designed to increase the census of these stars and to provide a chemical inventory that is as detailed as possible. Seven of the most iron-poor stars have been observed with the UVES spectrograph at the ESO VLT Kueyen 8.2m telescope to refine their chemical composition. We analysed the spectra based on 1D LTE model atmospheres, but also used 3D hydrodynamical simulations of stellar atmospheres. We measured carbon in six of the seven stars: all are carbon-enhanced and belong to the low-carbon band, defined in the TOPoS II paper. We measured lithium (A(Li)=1.9) in the most iron-poor star (SDSS J1035+0641, [Fe/H] < -5.2 ). We were also able to measure Li in three stars at [Fe/H]~ -4.0, two of which lie on the Spite plateau. We confirm that SDSS J1349+1407 is extremely rich in Mg, but not in Ca. It is also very rich in Na. Several of our stars are characterised by low alpha-to-iron ratios. The lack of high-carbon band stars at low metallicity can be understood in terms of evolutionary timescales of binary systems. The detection of Li in SDSS J1035+0641 places a strong constraint on theories that aim at solving the cosmological lithium problem. The Li abundance of the two warmer stars at [Fe/H]~ -4.0 places them on the Spite plateau, while the third, cooler star, lies below. We argue that this suggests that the temperature at which Li depletion begins increases with decreasing [Fe/H]. SDSS J1349+1407 may belong to a class of Mg-rich EMP stars. We cannot assess if there is a scatter in alpha-to-iron ratios among the EMP stars or if there are several discrete populations. However, the existence of stars with low alpha-to-iron ratios is supported by our observations.
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