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

组合学是研究数学结构的大小与其其他(几何/结构)性质之间关系的数学领域。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非常密切的联系；它也与代数、几何和数论有越来越多的联系。组合学中的一个经典问题是确定没有三角形的n-顶点图的最大可能边数；这个问题在一个多世纪前由Mantel解决，但类似的问题，即用一个长度为8的圈来代替三角形，至今仍未解决。对于超图，大多数类似的问题也是完全开放的。近年来，组合数学取得了许多令人振奋的进展，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。这个博士项目包括熟悉组合数学的研究水平的技术(包括那些利用代数、分析和概率方法的技术)，同时解决组合数学中一些未解决的问题。该项目的一个调查领域是鄂尔多斯-科-拉多型问题。它们要求物体家族中任何两个物体以某种方式‘一致’的最大可能大小。最近，几个Erdos-Ko-Rado型问题已被用代数和分析的技巧成功地解决了。然而，许多问题仍未得到解决。例如，SOS的一个问题：你可以取(1，2，...，n)的多少个子集，使得这些子集中的任何两个子集共享长度为3的算术级数？人们对这个问题几乎一无所知。另一个研究领域是等周不平等。等周问题是数学中的经典研究对象。一般而言，他们要求某一“大小”的物体尽可能小的“边界”。也许最古老的是平面上的等周问题：在区域1的平面的所有子集中，哪个边界最小？答案是古希腊人“知道的”，但直到19世纪才给出了严格的证明。在过去的五十年里，人们对“离散等周不等式”非常感兴趣。这些内容涉及图中边界的离散概念。它们在计算机科学和信息论中有重要的应用。这个领域中一个非常自然的未解决的问题是r-元素集的等周问题，这是由Bollobas和Leader推广的；还有许多其他问题。

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.