  A fundamental problem in pattern avoidance is describing the asymptotic
behavior of the extremal function and its generalizations. We prove an
equivalence between the asymptotics of the graph extremal function for a class
of bipartite graphs and the asymptotics of the matrix extremal function. We use
the equivalence to prove several new bounds on the extremal functions of
graphs. We develop a new method to bound the extremal function of hypergraphs
in terms of the extremal function of their associated multidimensional
matrices, improving the bound of the extremal function of $d$-permutation
hypergraphs of length $k$ from $O(n^{d-1})$ to $2^{O(k)}n^{d-1}$.
