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 Braß–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.

有序超图是顶点集线性有序的超图，凸几何超图是顶点集循环有序的超图。有序和凸几何图的极值问题具有丰富的历史，可应用于组合几何中的各种问题。在本文中，我们考虑均匀超图的类似极值问题，并确定各种有序和凸几何路径和匹配的极值函数的数量级。我们的结果概括了 Braß–Károlyi–Valtr、Capoyleas–Pach 和 Aronov–Dujmovič–Morin–Ooms-da Silveira 的早期工作。我们还在有序设置中提供了 Erdős-Ko-Rado 定理的新变体。