On supersaturation and stability for generalized Turán problems

Fix a graph (Formula presented.). We say that a graph is (Formula presented.) -free if it contains no copy of (Formula presented.) as a subgraph. Let (Formula presented.) denote the maximum number of copies of a graph (Formula presented.) in an (Formula presented.) -vertex (Formula presented.) -free graph. In this note, we will give a new general supersaturation result for (Formula presented.) in the case when (Formula presented.) as well as a new proof of a stability theorem for (Formula presented.).