Melting experiments on the Fe–C binary system up to 255 GPa: Constraints on the carbon content in the Earth's core

•Carbon concentration in the eutectic liquid is about 3 wt.% at ICB.•Fe forms eutectic melting with Fe3C to 203 GPa and with Fe7C3 at 255 GPa.•The solubility of carbon in solid Fe is almost constant at ∼1.0 wt.% above 20 GPa.•Neither Fe7C3 nor hcp Fe containing 1 wt.% C can explain the inner core density deficit.•The outer core liquid can be Fe–C–Si, Fe–C–S, or Fe–C–H.