Generalized Tur\'an results for disjoint cliques

Abstract: The generalized Tur'an number

m a t h r m e x (n, H, F)

is the largest number of copies of

H

in

n

-vertex

F

-free graphs. We denote by

t F

the vertex-disjoint union of

t

copies of

F

. Gerbner, Methuku and Vizer in 2019 determined the order of magnitude of

m a t h r m e x (n, K_{s}, t K_{r})

. We extend this result in three directions. First, we determine

m a t h r m e x (n, K_{s}, t K_{r})

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

m a t h r m e x (n, H, t K_{r})

for every graph

H

, and also of the analogous number for

p

-uniform hypergraphs.