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

Erdos-Ko-Rado 类型问题、等周不等式以及组合学中的其他主题。

• 批准号: 2614845
• 金额: --
• 依托单位: University of Bristol
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2021
• 资助国家: 英国
• 起止时间: 2021 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2614845
• 关键词: Erdos; Ko; Rado; type; problems

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.

组合学是数学领域，与数学结构的大小及其其他（几何/结构）特性之间的关系有关。它主要与离散的数学对象有关，例如图形和超图。它与理论计算机科学和离散分析有着密切的联系；它还与代数，几何和数理论具有不断增长的联系。 Comminatorics中问题的经典示例是确定N-Vertex图中没有三角形的最大边数。一个世纪前的曼特尔（Mantel）解决了这个问题，但是类似的问题是，一个人以八个循环为八的三角形替代了三角形，直到今天。大多数用于超图的类似问题，也保持完全开放。近年来，Combinatorics的进步取得了令人兴奋的进步，从组合本身以及来自代数，分析和概率理论等数学领域的其他领域都采用了技术。该博士学位项目涉及熟悉组合学中的研究级技术（包括利用代数，分析和概率方法的技术），同时在组合学中同时解决了一些未解决的问题。该项目中的一个调查领域是ERDOS-KO-RADO型问题。这些要求在某种程度上以某种方式“同意”的对象家族中最大的大小。最近，使用代数和分析的技术成功解决了几种ERDOS-KO-RADO型问题。但是，许多人仍未解决。例如，SOS的问题：您可以接受多少个（1，2，...，n）的子集，以便任何两个子集共享长度3的算术进程？几乎对这个问题一无所知。调查的另一个领域是等等不平等现象。等法问题是数学研究的经典对象。通常，他们要求某个“大小”对象的最小可能的“边界”。也许最古老的是平面中的等法问题：在区域1的所有子集中，边界最小？答案是古希腊人的“知道”，但直到19世纪才给出了严格的证据。在过去的五十年中，人们对“离散等级不平等”引起了极大的兴趣。这些涉及图中边界的离散概念。它们在计算机科学和信息理论中具有重要的应用。在该领域，一个非常自然的未解决问题是R元素集的等等问题，由Bollobas和Leader推广。还有很多。