We generalize techniques by Coskun, Riedl, and Yeong, and obtain an almost optimal bound on the degree for the algebraic hyperbolicity of very general hypersurfaces in rational homogeneous varieties. As examples, we work out the cases of very general hypersurfaces in Grassmannians and products therefore, orthogonal and symplectic Grassmannians, and flag varieties.

• 14J70 - Hypersurfaces and algebraic geometry
• 14M15 - Grassmannians, Schubert varieties, flag manifolds
• 14M17 - Homogeneous spaces and generalizations
• 32Q45 - Hyperbolic and Kobayashi hyperbolic manifolds