Generalized Turán results for disjoint cliques

Abstract

The generalized Turán number ex(n, H, F ) is the largest number of copies of H in nvertex 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, K s , tK r ). We extend this result in three directions. First, we determine ex(n, K s , tK 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 ex(n, H, tK r ) for every graph H, and also of the analogous number for p-uniform hypergraphs.

Generalized Turán problems for double stars

“…The order of magnitude of ex(n, H, M s+1 ) was determined in [9] for every graph H. Given a graph G and a set U ⊂ V (G), the partial (m, U)-blowup of G is obtained the following way. We replace each vertex u ∈ U with m vertices u 1 , .…”

“…Let us consider a largest set U ⊂ V (H) such that no partial (m, U)-blowup of H contains M s+1 , and let b(H) = b(H, s) denote the order of U. A theorem in [9] shows that ex(n,…”