极值理论是近几十年来组合学与图论研究领域中一直十分活跃的一个研究方向。极值的确定以及达到极值时结构的确定是这一领域的主要研究内容。有限集上的Erdős-Ko-Rado定理（简称EKR定理）是这一领域的核心定理，以往关于该定理的推广主要集中在子集系，有限向量空间以及置换群等具体的对象上。在本项目中我们将从简单图和超图等不同的角度推广EKR定理，把不同领域中看似孤立的问题通过EKR定理联系在一起，试图在一个一般的框架体系下研究极值问题。具体内容为：Kneser图的2-独立集和n-2r+1独立集研究；点传递图以及其张量积的独立数，分数色数研究，超图中关于边数与匹配数关系的Erdős猜想研究。力争解决其中的一些公开问题和猜想。相关结论的取得将会丰富EKR定理，促进组合极值理论的发展。

The extremal combinatorics is one of the most active research areas in combinatorics and graph theory. Typical problems are to determine the extremal value and structure of a certain systems. The Erdős-Ko-Rado theorem of finite set is one of the central theorems of this field. There are many generalizations on this theorem. Those generalizations are mainly concentrated on set systems, vector space of finite fields and symetric group, etc. In this project, we will generalize it from the aspects of graph and hypergraph, and develop a uniform framework for generalizing EKR theorem in different fields. Particular problems we shall study in this project include the 2-independent set of kneser graphs, the independence number and the fractional chromatic number of the tensor product of three graphs, the Erdós conjecture on the relation between matching number and edge number of hypergraphs. We shall try to solve some open problems and conjectures.