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

Authors:
Martin Klazar
Affiliations:
Department of Applied Mathematics (KAM) and Institute for Theoretical Computer Science (ITI), Charles University, Malostranské námestí 25, 118 00 Praha, Czech Republic
Venue:
European Journal of Combinatorics
Year:
2004

Citing 7
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Abstract

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 n log 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.