Turán型问题是极值组合中的重要问题之一。拉格朗日方法是解决超图Turán问题的重要工具。2013年，Hefetz和Keevash提出了确定一般超图的拉格朗日密度问题。任何一个r一致超图的拉格朗日密度对应了另一个r一致超图(扩张)的Turán密度。例如，求四面体的Turán密度等价于求其拉格朗日密度。然而，不论是Turán问题，还是拉格朗日密度问题，目前已知的结果都非常少。本项目以求超图的拉格朗日密度和Turán数为目标，通过结构分析、左压缩操作和超图邻域等方法，研究r一致超图的拉格朗日密度，再结合稳定性方法研究相应超图的Turán数和极值结构。项目的开展将有利于丰富Turán型问题的研究内容和方法，进一步促进Turán型问题在国内的传播和发展。

Turán type problems are central to the development of extremal combinatorics. There are very few known results about the Turán number of a hypergraph. Lagrangian method has been an important tool in Turán type problems. Lagrangian density is closely related to Turán dnesity. Exploring the Lagrangian density of a hypergraph itself is an interesting and challenging question. Indeed, to answer a long standing question of Turán on the Turán number of the Tetrahedron is equivalent to determine the Lagrangian density of the Tetrahedron. The goal of this project is to solve some Turán numbers of hypergraphs. Combining hypergraph structural analysis, left-compressed technique, hypergraph link, and local Lagangian method, we study the Lagangian densities of r-uniform hypergraphs. Using the stability method, we intend to solve some Turán numbers of r-uniform hypergraphs. Launching of the project will enrich the research content and method of Turán type problem, further improve the domestic communication and development of Turán type problem.