Let $F$ be a graph. We say that a hypergraph $H$ is a { \it Berge}-$F$ if there is a bijection $f : E(F) \rightarrow E(H )$ such that $e \subseteq f(e)$ for every $e \in E(F)$. Note that Berge-$F$ actually denotes a class of hypergraphs. The maximum number of edges in an $n$-vertex $r$-graph with no subhypergraph isomorphic to any Berge-$F$ is denoted $\ex_r(n,\textrm{ Berge-}F)$. In this paper we establish new upper and lower bounds on $\ex_r(n,\textrm{ Berge-}F)$ for general graphs $F$, and investigate connections between $\ex_r(n,\textrm{ Berge-}F)$ and other recently studied extremal functions for graphs and hypergraphs. One case of specific interest will be when $F = K_{ s,t}$. Additionally, we prove a counting result for $r$-graphs of girth five that complements the asymptotic formula $\textup{ ex}_3 (n , \textrm{ Berge-}\{ C_2 , C_3 , C_4 \} ) = \frac{ 1}{ 6} n^{ 3/2} + o( n^{ 3/2} )$ of Lazebnik and Verstra\"{ e}te [{ \em Electron.\ J. of Combin}. { \bf 10}, (2003)].
Categories: math.CO