On supersaturation and stability for generalized Tur\'an problems

Fix graphs $F$ and $H$. Let $\mathrm{e}\mathrm{x}(n,H,F)$ denote the maximum number of copies of a graph $H$ in an $n$-vertex $F$-free graph. In this note we will give a new general supersaturation result for $\mathrm{e}\mathrm{x}(n,H,F)$ in the case when $\chi (H)<\chi (F)$ as well as a new proof of a stability theorem for $\mathrm{e}\mathrm{x}(n,K_r,F)$.