Extremal problems for ordered (hyper)graphs: applications of Davenport–Schinzel sequences

We introduce a containment relation of hypergraphs which respects linear orderings of vertices, and we investigate associated extremal functions. We extend, using a more generally applicable theorem, the nlog n upper bound on sizes of ({1,3},{1,5},{2,3},{2,4})-free ordered graphs with n vertices, due to Füredi, to the n(log n)2(log log n)3 upper bound in the hypergraph case. We apply Davenport–Schinzel sequences and obtain almost linear upper bounds in terms of the inverse Ackermann function α(n). For example, we obtain such bounds in the case of extremal functions of forests consisting of stars all of whose centres precede all leaves.