Layered, hexagonal crystal structures, like zeta and eta phases, play an important role in ultra-high temperature ceramics, often significantly increasing toughness of carbide composites. Despite their importance, open questions remain about their structure, stability, and compositional pervasiveness. We use high-throughput density functional theory to characterize the thermodynamic stability and elastic constants of layered carbides and nitrides M\(_{n+1}\)X\(_{n}\) with n = 1, 2, and 3, M = Ta, Ti, Hf, Zr, Nb, Mo, V, W, Sc, Cr, Mn and X = C, N. The explored set of vacancy-ordered transition metal carbide systems considers multiple stacking sequences of M\(_{n+1}\)X\(_{n}\)-type layers, as well as additional ordered strings of vacancies within the M\(_{n+1}\)X\(_{n}\) layers. We identified 74 new hexagonal, layered materials with thermodynamic stability comparable or better than previously observed zeta phases. To assess their potential for high temperature applications, we used machine learning and physics-based models with DFT inputs to predict their melting temperatures and discovered several candidates on par with the current state of the art zeta-like phases and four with predicted melting temperatures above 2500 K that lie on the convex hull. These findings expand the range of chemistries and structures for high-temperature applications.
