Erdos-Ko-Rado type problems, Isoperimetric inequalities, and other topics in Combinatorics.

组合数学（Combinatorics）是数学领域，关注数学结构的大小与其其他（几何/结构）属性之间的关系。它主要涉及离散的数学对象，如图和超图。它与理论计算机科学和离散分析有着非常密切的联系;它也与代数，几何和数论有着越来越多的联系。组合数学中一个经典的问题是确定一个没有三角形的n顶点图中最大可能的边数;这个问题在世纪前由Mantel解决，但是类似的问题，其中一个用长度为8的循环替换三角形，直到今天仍然开放。超图的大多数类似问题也仍然是完全开放的。近年来，组合数学取得了令人兴奋的进展，利用了组合数学本身以及其他数学领域（如代数，分析和概率论）的技术。这个博士项目涉及熟悉组合学的研究水平技术（包括那些利用代数，分析和概率方法），并同时解决组合学中一些未解决的问题。该项目的一个调查领域是Erdos-Ko-Rado型问题。这些要求一个对象族的最大可能的大小，其中任何两个对象在某种程度上“同意”。最近，几个Erdos-Ko-Rado类型的问题已经成功地解决了使用代数和分析的技术。然而，许多问题仍未解决。例如，Sos的问题：（1，2，.，n）你能取，使得任何两个子集共享一个长度为3的算术级数吗？实际上，对这个问题一无所知。另一个研究领域是等周不等式。等周问题是数学中的经典研究对象。一般来说，它们要求某一“大小”物体的尽可能小的“边界”。也许最古老的是平面上的等周问题：在面积为1的平面的所有子集中，哪一个具有最小的边界？古希腊人“知道”这个问题的答案，但直到世纪才给出了严格的证明。在过去的50年里，人们对离散等周不等式产生了极大的兴趣。这些处理离散概念的边界图。它们在计算机科学和信息理论中有重要的应用。一个非常自然的未解决的问题，在这方面是等周问题的r-元素集，普及的Bollobas和领导人;还有许多其他人。

Combinatorics is the area of mathematics that is concerned with the relationship between the size of mathematical structures, and their other (geometric/structural) properties. It is mainly concerned with discrete mathematical objects, such as graphs and hypergraphs. It has very close links with Theoretical Computer Science and Discrete Analysis; it also has growing connections to Algebra, Geometry and Number Theory. A classical example of a problem in Combinatorics is to determine the maximum possible number of edges in an n-vertex graph with no triangle; this problem was solved by Mantel over a century ago, but the analogous problem where one replaces a triangle with a cycle of length eight, remains open to this day. Most of the analogous problems for hypergraphs, also remain completely open. There has been much exciting progress in Combinatorics in recent years, utilising techniques both from within Combinatorics itself, and also from other areas of mathematics such as Algebra, Analysis and Probability Theory. This PhD project involves gaining familiarity with research-level techniques in Combinatorics (including those utilising algebraic, analytic and probabilistic methods), and simultaneously tackling some unsolved problems in Combinatorics. One area of investigation in the project is that of Erdos-Ko-Rado type problems. These ask for the largest possible size of a family of objects in which any two of the objects `agree' in some way. Recently, several Erdos-Ko-Rado type problems have been tackled successfully using techniques from Algebra and Analysis. Many, however, remain unsolved. For example, a question of Sos: how many subsets of (1, 2, ..., n) can you take, such that any two of the subsets share an arithmetic progression of length 3? Virtually nothing is known about this question. Another area of investigation is that of isoperimetric inequalities. Isoperimetric problems are classical objects of study in mathematics. In general, they ask for the smallest possible `boundary' of an object of a certain `size'. Perhaps the oldest is the isoperimetric problem in the plane: among all subsets of the plane of area 1, which has the smallest boundary? The answer was `known' to the ancient Greeks, but it was not until the 19th century that a rigorous proof was given. In the last fifty years, there has been a great deal of interest in `discrete isoperimetric inequalities'. These deal with discrete notions of boundary in graphs. They have important applications in computer science and information theory. One very natural unsolved problem in this area is the isoperimetric problem for r-element sets, popularised by Bollobas and Leader; there are many others.