On Non-degenerate Berge–Turán Problems

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