Extremal graph theory, Ramsey theory and additive combinatorics

在我博士的第一年，我致力于组合学(也称为离散数学)的三个主题。其中之一是极值图论，它涉及确定不包含给定子图的图的最大边数。第二种是拉姆齐理论。粗略地说，这个领域专注于在图或它的补图中寻找结构。最后，第三个领域是加性组合学，它研究群的子集。(组是可以定义加法的对象，例如整数或实数。)最后一个项目在理论计算机科学方面也具有重要意义。与我的主管Timothy Gowers一起，我们成功地在所有这三个领域建立了可发表的结果。在这些项目中广泛使用的一种常用技术是所谓的概率方法，它是一种强大的工具，允许我们通过从所有可用对象中随机选择一个对象来发现具有特定属性的对象。在未来，我将致力于解决上述三个领域中的进一步开放问题。

In the first year of my PhD I was working on three topics in combinatorics (also known as discrete mathematics). One of them is extremal graph theory which is concerned with determining the maximal number of edges in a graph which does not contain a given subgraph. The second one is Ramsey theory. Roughly speaking, this field focuses on finding structures in a graph or its complement. Finally, the third area is additive combinatorics, which investigates subsets of groups. (Groups are objects on which addition can be defined, such as the integers or real numbers.) This last project also has significance in theoretical computer science.Together with my supervisor, Timothy Gowers, we have managed to establish publishable results in all three of these areas. A common technique which was used heavily in these projects is the so-called probabilistic method, which is a powerful tool allowing us to find objects with certain properties by randomly picking one object of all the available ones.In the future I will be aiming to solve further open problems in the three areas mentioned above.