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Motivated by recent experiments on ABC-stacked rhombohedral trilayer graphene(RTG) which observed spin-valley symmetry-breaking and superconductivity, westudy instabilities of the RTG metallic state to symmetry breaking orders. Wefind that interactions select the inter-valley coherent order (IVC) as thepreferred ordering channel over a wide range, whose theoretically determinedphase boundaries agree well with experiments on both the hole and electrondoped sides. The Fermi surfaces near van Hove singularities admit partialnesting between valleys, which promotes both inter-valley superconductivity andIVC fluctuations. We investigate the interplay between these fluctuations andthe Hunds (intervalley spin) interaction using a renormalization groupapproach. For antiferromagnetic Hund's coupling, intervalley pairing appears inthe spin-singlet channel with enhanced $T_c$, that scales with thedimensionless coupling $g$ as $T_c\sim\exp(-1/\sqrt{g})$ , compared to thestandard $\exp(-1/g)$ scaling. In its simplest form, this scenario assumes asign change in the Hund's coupling on increasing hole doping. On the otherhand, the calculation incorporates breaking of the independent spin rotationsbetween valleys from the start, and strongly selects spin singlet over spintriplet pairing, and naturally occurs in proximity to the IVC, consistent withobservations.

Kohn-Luttinger Superconductivity and Inter-Valley Coherence in Rhombohedral Trilayer Graphene