Generalized Turán results for disjoint cliques

The generalized Turán number ex(n,H,F) is the largest number of copies of H in n-vertex F-free graphs. We denote by tF the vertex-disjoint union of t copies of F. Gerbner, Methuku and Vizer in 2019 determined the order of magnitude of ex(n,Ks,tKr). We extend this result in three directions. First, we determine ex(n,Ks,tKr) exactly for sufficiently large n. Second, we determine the asymptotics of the analogous number for p-uniform hypergraphs. Third, we determine the order of magnitude of ex(n,H,tKr) for every graph H, and also of the analogous number for p-uniform hypergraphs.