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 investigate associated extremal functions. We extend, by means of a more generally applicable theorem, the n log n upper bound on the ordered graph extremal function of F = ({1, 3}, {1, 5}, {2, 3}, {2, 4}) due to Füredi to the n(log n)2(log log n)3 upper bound in the hypergraph case. We use Davenport–Schinzel sequences to derive almost linear upper bounds in terms of the inverse Ackermann function α(n). We obtain such upper bounds for the extremal functions of forests consisting of stars whose all centers precede all leaves.