Explore higher-order generalizations of stability and arithmetic regularity in this 46-minute lecture by Caroline Terry from The Ohio State University. Delve into the concept of regularity lemmas and their applications in higher arities. Examine stability in the context of hereditary graph properties and learn about its characterization. Investigate 3-graphs, their definitions, and regularity principles. Discover methods to constrain irregular triads and understand the characterization of linear error. Analyze the arithmetic setting, quadratic factors, and the structure of NFOP sets. Gain insights into essential tools and explore further directions in this field of study. This talk is part of the "From Geometric Stability Theory to Tame Geometry" workshop at the Fields Institute.