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）属性之间关系的数学领域。它主要关注离散数学对象，如图和超图。它与理论计算机科学和离散分析有着密切的联系；它与代数、几何和数论的联系也越来越紧密。组合学中一个经典的例子是，在一个没有三角形的n顶点图中确定最大可能的边数；这个问题在一个多世纪前就被曼特尔解决了，但类似的问题——用一个长度为8的圆来代替三角形——直到今天仍然没有解决。对于超图，大多数类似的问题仍然是完全开放的。近年来，组合学取得了许多令人兴奋的进展，这些进展既利用了组合学本身的技术，也利用了代数、分析和概率论等其他数学领域的技术。该博士项目涉及熟悉组合学的研究级技术（包括利用代数、分析和概率方法的技术），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。这些问题要求在任意两个物体在某种程度上“一致”的情况下，求得一组物体的最大可能大小。最近，一些Erdos-Ko-Rado类型的问题已经成功地利用代数和分析的技术解决了。然而，许多问题仍未得到解决。例如，一个Sos问题：（1,2，…）有多少个子集， n)可以取任意两个子集共享长度为3的等差数列吗？实际上，关于这个问题我们一无所知。另一个研究领域是等周不等式。等周问题是数学中经典的研究对象。一般来说，它们要求具有一定“尺寸”的物体的最小可能“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，哪个子集的边界最小？古希腊人就“知道”这个问题的答案，但直到19世纪才有了严谨的证明。在过去的五十年里，人们对“离散等周不等式”产生了极大的兴趣。它们处理图中边界的离散概念。它们在计算机科学和信息理论中有着重要的应用。在这个领域中，每个自然未解的问题都是由Bollobas和Leader推广的r元素集的等周问题；摘要（不超过4000个字符，包括空格，清楚地说明该项目涉及EPSRC的研究领域-更多信息见背面）组合学是数学领域，关注数学结构的大小与其其他（几何/结构）属性之间的关系。它主要关注离散数学对象，如图和超图。它与理论计算机科学和离散分析有着密切的联系；它与代数、几何和数论的联系也越来越紧密。组合学中一个经典的例子是，在一个没有三角形的n顶点图中确定最大可能的边数；这个问题在一个多世纪前就被曼特尔解决了，但类似的问题——用一个长度为8的圆来代替三角形——直到今天仍然没有解决。对于超图，大多数类似的问题仍然是完全开放的。近年来，组合学取得了许多令人兴奋的进展，这些进展既利用了组合学内部的技术，也利用了其他技术

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 hasgrowing 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. Onevery natural unsolved problem in this area is the isoperimetric problem for r-element sets, popularised by Bollobas and Leader; there areAbstract(no more than 4,000 characters including spaces, clearly explain which EPSRC research area the project relates to - for more info see overleaf) 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 hasgrowing 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