Extremal Combinatorics

这个项目集中于极值组合学的主题，研究离散系统中有用参数之间的关系，并在这些系统的各种族上表征它们的极值。这些问题通常在计算机科学和其他数学领域有应用，但它们本身也很优雅和有趣。这个项目的目标是进一步开发可用的工具箱来解决组合问题。这一目标将通过研究某些问题族来实现，这些问题族可以根据其解决方案中使用的方法进行组织。首先，我们研究了解析（连续）参数在离散问题中的应用。其次，我们研究了集系统的极值问题。最后，我们探讨了概率方法在纯确定性问题中的应用。组合学领域包括对离散结构和过程的严格数学研究，如集合、网络甚至算法。过去，组合问题通常是靠纯粹的聪明才智来解决的。然而，今天出现了各种强大的方法，它们吸取了数学科学许多其他分支的元素。与此同时，计算机科学的快速发展大大增加了对离散系统基础结果的需求——这些可以为未来的工作提供理论基础。这个项目的首要目标是使用问题驱动的哲学来激发组合学新技术发展的创新。统一本研究中的问题是一个简单而优雅的主题，它激发了来自整个数学谱的方法。因此，这些问题也为年轻人参与研究提供了很好的平台。

This project focuses on topics in Extremal Combinatorics, which investigate the relationships between useful parameters of discrete systems, and characterizes their extreme values over various families of those systems. Such problems often have applications in Computer Science and other areas of Mathematics, but are also elegant and interesting in their own right. The goal of this project is to further develop the toolbox of available approaches for attacking Combinatorial problems. This goal will be achieved by studying certain families of problems which can be organized according to the methods used in their solutions. First, we examine applications of analytical (continuous) arguments to discrete problems. Second, we study extremal problems related to set systems. Finally, we explore the application of probabilistic methods to purely deterministic problems.The field of combinatorics encompasses the rigorous mathematical study of discrete structures and processes, such as sets, networks, and even algorithms. In the past, combinatorial problems were often solved by pure ingenuity. Today, however, a variety of powerful methods have emerged, which draw elements from many other branches of the mathematical sciences. At the same time, the rapid development of Computer Science has substantially increased the demand for foundational results on discrete systems--these can provide the theoretical basis for future work. The overarching objective of this project is to use a problem-driven philosophy to inspire innovations in the development of new techniques in Combinatorics. Unifying the problems in this study is a theme of simple, elegant statements that inspire approaches which arise from across the mathematical spectrum. Consequently, these problems also serve well as a platform for bringing young people into research.