Initial-final mass relation for single stars. Models with (M10) and without (M1) pair-instability pulsation supernovae and pair-instability supernovae are shown. Bottom panel: at high metallicity (Z = 10% Z_⊙ and higher) the models are indistinguishable. For the border-line metallicity of Z = 10% Z_⊙, stars with very high initial mass (M_zams> 100 M_⊙) will form slightly lighter BHs (by ~ 2 M_⊙) if pair-instability pulsation supernovae are included. Top panel: at low metallicity (e.g., Z = 0.5% Z_⊙) pair-instability pulsation supernovae and pair-instability supernovae do not allow for high mass BH formation; maximum BH mass is M_BH = 45 M_⊙. For the very low metallicity of Z = 0.5% Z_⊙, very massive stars (M_zams ≈ 100−140 M_⊙) lose significant mass in pair-instability pulsation supernovae reducing the BH mass to M_BH ≈ 40 M_⊙, while the most massive stars (M_zams> 140 M_⊙) explode in pair-instability supernovae leaving no remnant.