Extremal theory of vertex or edge ordered graphs

We enrich the structure of finite simple graphs with a linear order on either the vertices or the edges. Extending the standard question of Turan-type extrema' graph theory we ask for the maximal number of edges in such a vertex or edge ordered graph on n vertices that does not contain a given pattern (or several patterns) as a subgraph. The forbidden subgraph itself is also a vertex or edge ordered graph, so we forbid a certain subgraph with a specified ordering, but we allow the same underlying subgraph with a different (vertex or edge) order. This allows us to study a large number of extrema' problems that are not expressible in the classical theory. In this survey we report on ongoing research. For easier access we include sketches of proofs of selected results.