Context. The old Galactic halo stars hold the fossil record of the interstellar medium chemical composition at the time of their formation. Most of the stars studied so far are relatively near to the Sun, this prompts the study of more distant stars, both to increase the size of the sample and to search for possible variations of abundance patterns at greater distances. Aims: The purpose of our study is to determine the chemical composition of a sample of 16 candidate extremely metal-poor (EMP) dwarf stars, extracted from the Sloan Digital Sky Survey (SDSS). There are two main purposes: in the first place to verify the reliability of the metallicity estimates derived from the SDSS spectra; in the second place to see if the abundance trends found for the brighter nearer stars studied previously also hold for this sample of fainter, more distant stars. Methods: We used the UVES at the VLT to obtain high-resolution spectra of the programme stars. The abundances were determined by an automatic analysis with the MyGIsFOS code, with the exception of lithium, for which the abundances were determined from the measured equivalent widths of the Li i resonance doublet. Results: All candidates are confirmed to be EMP stars, with [Fe/H] ≤ -3.0. The chemical composition of the sample of stars is similar to that of brighter and nearer samples. We measured the lithium abundance for 12 stars and provide stringent upper limits for three other stars, for a fourth star the upper limit is not significant, owing to the low signal-to noise ratio of the spectrum. The "meltdown" of the Spite plateau is confirmed, but some of the lowest metallicity stars of the sample lie on the plateau. Conclusions: The concordance of the metallicities derived from high-resolution spectra and those estimated from the SDSS spectra suggests that the latter may be used to study the metallicity distribution of the halo. The abundance pattern suggests that the halo was well mixed for all probed metallicities and distances. The fact that at the lowest metallicities we find stars on the Spite plateau suggests that the meltdown depends on at least another parameter, besides metallicity. Based on spectra obtained with UVES at the 8.2 m Kueyen ESO telescope, programmes 078.D-0217 and 081.D.0373.Table 1 is available in electronic form at http://www.aanda.org