Published online by Cambridge University Press: 10 August 2020
An ordered hypergraph is a hypergraph whose vertex set is linearly ordered, and a convex geometric hypergraph is a hypergraph whose vertex set is cyclically ordered. Extremal problems for ordered and convex geometric graphs have a rich history with applications to a variety of problems in combinatorial geometry. In this paper, we consider analogous extremal problems for uniform hypergraphs, and determine the order of magnitude of the extremal function for various ordered and convex geometric paths and matchings. Our results generalize earlier works of Braı–Károlyi–Valtr, Capoyleas–Pach, and Aronov–Dujmovič–Morin–Ooms-da Silveira. We also provide a new variation of the Erdős-Ko-Rado theorem in the ordered setting.
Research supported by grant K116769 from the National Research, Development and Innovation Office NKFIH and by the Simons Foundation Collaboration grant #317487. Research partially supported by NSF grants DMS-1400249, DMS-1855542, DMS-1300138, DMS-1763317, and DMS-1556524, as well as DMS-1600592 by Award RB17164 of the UIUC Campus Research Board and by grants 18-01-00353A and 19-01-00682 of the Russian Foundation for Basic Research.