The generalized Tur\'an 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.Comment: 10 page