The maximum number of copies of any hypergraph H in Berge-K_k-free r-graphs is achieved by the balanced complete (k-1)-partite r-graph for all sufficiently large k, with related bounds connecting hypergraph and graph cases.
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For any tree T and family F, ex(n,T,F) is either Ω(n^{k+1}) or O(ex(n,F)^k) for every integer k.
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Generalized Tur\'an problems for Berge hypergraphs
The maximum number of copies of any hypergraph H in Berge-K_k-free r-graphs is achieved by the balanced complete (k-1)-partite r-graph for all sufficiently large k, with related bounds connecting hypergraph and graph cases.
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Helly Theorems for Generalized Tur\'an Problems
For any tree T and family F, ex(n,T,F) is either Ω(n^{k+1}) or O(ex(n,F)^k) for every integer k.