极值图论是组合图论中的核心研究内容之一，由于结果的深刻性以及从中发展出来的理论方法使得极值图论成为一个重要的研究领域，其在编码理论、离散几何和计算机科学等领域有广泛的应用。本课题将研究带有禁用子图结构的Turán类型极值问题,以图的扩张为禁用子超图的超图极值问题和超图的p-谱半径的极值等问题，给出这些问题的极值数并刻画相应的极图。我们将围绕著名学者Erdős关于匹配扩张的极值数猜想、Kostochka关于树宽不超过2的图的扩张、无圈3-染色平面图扩张的极值数的两个公开问题和Nikiforov提出的关于一致超图的谱半径下界的公开问题等开展工作，运用组合构造方法、优化方法、代数方法、随机方法和张量理论等对这些问题进行研究。研究内容涉及图和超图的横贯、匹配、染色、谱半径等重要结构参数。该项目的研究目标是对这些问题和猜想有实质性的推进,同时我们也将探索一些新的理论和方法。

Extremal graph theory is one of core research fields in combinatorics and graph theory. Due to the profound results and relevant methods, the research on extremal problems has attracted much attention and it becomes one of the most important topics in combinatorics. Extremal graph theory has many applications in coding theory, discrete geometry, theoretical computer science. In this project, we will discuss extremal problems of Turán type of graphs, extremal problems of Turán type of hypergraphs, extremal problems for p-spectral radius of hypergraphs. We will focus on matching conjecture proposed by Erdős, open problems on extremal function of expansion of a graph with tree-width at most 2, extremal function of expansion of a planar graph with an acyclic 3-coloring proposed by Kostochka and an open problem on lower bound of spectral radius of r-uniform hypergraphs proposed by Nikiforov. By using analytic methods, algebraic methods, probabilistic methods and tensor theory, we shall work on these problems. This project involves in transversal, matching, coloring and spectral of graphs and hypergraphs. Our aim is to solve or partly solve these problems. Meanwhile, we will find some new and effective methods and theories.