Extremal and probabilistic combinatorics

这个项目位于极值和概率组合数学领域，并将解决该领域感兴趣的热门问题。这些问题包括：(1)拟随机超图的Dirac问题。有一个成熟的准随机图理论和条件，允许在这些图中嵌入大个子图。最近，超图中的拟随机性建立了一个更一般的分类，随之而来的是一系列有趣的公开问题，涉及到将嵌入结果推广到超图的情况。(2)编辑图形和超图中的距离。这是关于图和超图的一个研究得很好的度量，对于这些图和超图，例如，关于从给定性质到图或超图(可能具有固定密度)的最大编辑距离，仍然有重要的未决问题。(3)在竞赛图中嵌入无圈定向图。最近的几个结果讨论了树在竞赛中的嵌入，一般的问题是对给定的树T找到最小的n，使得n个顶点上的每个竞赛都包含一个T的副本。这个问题通过用不含有向圈的有向图来代替‘树’来自然地推广，但到目前为止对这种情况知之甚少。这些主题都是基本问题，它们的各个方面都引起了世界各地领先研究人员的关注。

This project lies in the area of extremal and probabilistic combinatorics, and will address topical questions of interest in this field. These may include: (1) Dirac-type problems for quasi-random hypergraphs. There is a well-developed theory of quasi-random graphs and conditions which allow large subgraphs to be embedded within these graphs. Recently a more general classification of quasirandomness in hypergraphs has been established, bringing with it a range of interesting open problems regarding generalisations of embedding results to the hypergraph case. (2) Edit distances in graphs and hypergraphs. This is a well-studied metric on graphs and hypergraphs, for which there are still important open questions regarding, for example, the maximum edit distance of a graph or hypergraph (possibly of fixed density) from a given property. (3) Embeddings acyclic oriented graphs in tournaments. Several recent results have addressed embeddings of trees in tournaments, with the general question being to find for a given tree T the smallest n such that every tournament on n vertices contains a copy of T. This question generalises naturally by replacing `trees' with `oriented graphs not containing directed cycles', but little is known about this case so far. These topics are fundamental questions, aspects of which have attracted the attention of leading researchers around the world.