In the course of the TOPoS (Turn Off Primordial Stars) survey, aimed at discovering the lowest metallicity stars, we have found several carbon-enhanced metal-poor (CEMP) stars. We here present our analysis of six CEMP stars. Calcium and carbon are the only elements that can be measured in all six stars. The range is -5.0<=[Ca/H]< -2.1 and 7.12<=A(C)<=8.65. For star SDSS J1742+2531 we were able to detect three FeI lines from which we deduced [Fe/H]=-4.80, from four CaII lines we derived [Ca/H]=-4.56, and from synthesis of the G-band we derived A(C)=7.26. For SDSS J1035+0641 we were not able to detect any iron lines, yet we could place a robust (3sigma) upper limit of [Fe/H]< -5.0 and measure the Ca abundance, with [Ca/H]=-5.0, and carbon, A(C)=6.90. No lithium is detected in the spectrum of SDSS J1742+2531 or SDSS J1035+0641, which implies a robust upper limit of A(Li)<1.8 for both stars. Our measured carbon abundances confirm the bimodal distribution of carbon in CEMP stars, identifying a high-carbon band and a low-carbon band. We propose an interpretation of this bimodality according to which the stars on the high-carbon band are the result of mass transfer from an AGB companion, while the stars on the low-carbon band are genuine fossil records of a gas cloud that has also been enriched by a faint supernova (SN) providing carbon and the lighter elements. (Abridged)
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