General lemmas for Berge-Tur\'an hypergraph problems

For a graph $F$, a hypergraph $\mathcal{H}$ is a Berge copy of $F$ (or a Berge-$F$ in short), if there is a bijection $f : E(F) \rightarrow E(\mathcal{H})$ such that for each $e \in E(F)$ we have $e \subset f(e)$. A hypergraph is Berge-$F$-free if it does not contain a Berge copy of $F$. We denote the maximum number of hyperedges in an $n$-vertex $r$-uniform Berge-$F$-free hypergraph by $\mathrm{ex}_r(n,\textrm{Berge-}F).$ In this paper we prove two general lemmas concerning the maximum size of a Berge-$F$-free hypergraph and use them to establish new results and improve several old results. In particular, we give bounds on $\mathrm{ex}_r(n,\textrm{Berge-}F)$ when $F$ is a path (reproving a result of Gy\H{o}ri, Katona and Lemons), a cycle (extending a result of F\"uredi and \"Ozkahya), a theta graph (improving a result of He and Tait), or a $K_{2,t}$ (extending a result of Gerbner, Methuku and Vizer). We also establish new bounds when $F$ is a clique (which implies extensions of results by Maherani and Shahsiah and by Gy\'arf\'as) and when $F$ is a general tree.