On Non-degenerate Berge–Turán Problems

Given a hypergraph \({{\mathcal {H}}}\) and a graph G, we say that \({{\mathcal {H}}}\) is a Berge-G if there is a bijection between the hyperedges of \({{\mathcal {H}}}\) and the edges of G such that each hyperedge contains its image. We denote by \(\textrm{ex}_k(n,Berge- F)\) the largest number of hyperedges in a k-uniform Berge-F-free graph. Let \(\textrm{ex}(n,H,F)\) denote the largest number of copies of H in n-vertex F-free graphs. It is known that \(\textrm{ex}(n,K_k,F)\le \textrm{ex}_k(n,Berge- F)\le \textrm{ex}(n,K_k,F)+\textrm{ex}(n,F)\), thus if \(\chi (F)>r\), then \(\textrm{ex}_k(n,Berge- F)=(1+o(1)) \textrm{ex}(n,K_k,F)\). We conjecture that \(\textrm{ex}_k(n,Berge- F)=\textrm{ex}(n,K_k,F)\) in this case. We prove this conjecture in several instances, including the cases \(k=3\) and \(k=4\). We prove the general bound \(\textrm{ex}_k(n,Berge- F)= \textrm{ex}(n,K_k,F)+O(1)\).