Extremal Combinatorics

极值组合学研究离散结构中各种参数之间的关系。这一领域在过去几十年中经历了显著的增长。计算机科学和运筹学的各个方面受到大规模实际问题的推动，越来越依赖于复杂的组合技术，并在离散数学中提出了一系列新的具有挑战性的问题。与此同时，强大而深入的数学方法的发展极大地扩展了现在可以处理的组合问题的视野，满足了上述许多挑战。该项目将集中在极值组合学的中心问题。两个例子是Turan函数，它询问局部限制如何影响超图的全局大小，以及Ramsey理论，它研究大型结构是否包含高度有序的子结构。这些问题似乎是出了名的困难，甚至一些基本的问题仍然悬而未决。以前的尝试，虽然没有完全成功，导致了一些有用的技术和见解。最近的一些发展（如超图的正则性，图的限制，和标志代数）给我们提供了新的强大的工具，可能有助于在这些问题上取得进展。该项目旨在更好地了解这些领域，并开发普遍有用的方法和技术。

Extremal Combinatorics studies relations between various parameters of discrete structures. This area experienced a remarkable growth in the last few decades. Various aspects of Computer Science and Operations Research motivated by large-scale practical problems have been relying on more and more sophisticated combinatorial techniques and have posed a whole array of new challenging problems in Discrete Mathematics. At the same time, the development of powerful and deep mathematical methods has greatly expanded the horizon of combinatorial questions that can be approached now, meeting many of the above challenges. The project will concentrate on central questions of Extremal Combinatorics. Two examples are the the Turan function that asks how local restrictions can affect the global size of a hypergraph and the Ramsey theory that investigates whether large structures contain highly ordered substructures. These problems seem to be notoriously difficult and even some basic questions remain open. The previous attempts, although not completely successful, led to a number of useful techniques and insights. Some recent developments (such as hypergraph regularity, graph limits, and flag algebras) give us new powerful tools that may be instrumental in obtaining progress on these problems. The project aims at achieving a better understanding of these areas and developing generally useful methods and techniques.