General lemmas for Berge Turan 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 Gyori, Katona and Lemons) a cycle (extending a result of Furedi 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 Gyarfas) and when $F$ is a general tree.
Eur. J. Comb.