Turán型问题是极值组合中的核心问题与经典难题,在对Turán型问题的研究中一个重要的方法是超图的拉格朗日方法。Sidorenko证明了一致超图F的Turán密度等于所有Dense的不能将F同态嵌入的一致超图拉格朗日的上确界。将Sidorenko的结果应用到Turán密度中时，有两个关键问题，一是对Dense超图的刻画，或者更特殊地，对不能将F同态嵌入Dense的一致超图性质的刻画，二是对相关超图的拉格朗日的估算。本项目将对这两个问题进行研究，进而应用到对超图的Turán密度的估计中，并利用超图的正则性和Removal引理讨论稳定性从而得到某些超图Turán数的精确值。受拉格朗日方法的启发，本项目将更一般地研究如何构造由超图决定的连续函数，建立其最优值与超图最大团的关系从而估算超图的Turán密度。对这些问题的研究在计算机科学如复杂性分析，图像处理，数据处理等中也有应用。

Turán type problems are central to the development of extremal combinatorics. Lagrangian method is an important tool in this type of questions. In 1980’s, Sidorenko showed that the Turán density of an uniform hypergraph F equals the supremum of the Lagrangians of all dense F-homomorphism free uniform hypergraphs. To apply Sidorenko’s results, we need to understand two key questions. One is to understand dense uniform hypergrahs, or specifically, describe all dense F-homomorphism free uniform hypergraphs, another is to estimate their Lagrangians. In this project, we will explore these two questions and apply the results to give hypergraph Turán densities, then obtain the exact value of Turán numbers of some hypergraphs via the discussion on stability. In general, we will explore how to construct a good continuous function determined by a hypergraph, then build the connection of the optimum value of this continuous function and the maximum cliques of a hypergraph, and apply the connection in estimating Turán density of a hypergraph. Research on this type of questions is not only important from theoretical point of view, it has also important applications in computer science such as computational complexity and imaging processing.