几类3一致超图的拉格朗日密度和Turán密度

Turán型问题是极值图论中的核心问题之一。超图的拉格朗日是研究超图Turán型问题的重要方法。Sidorenko证明了一致超图的拉格朗日密度等于其扩张的Turán密度，特别地，若一致超图覆盖顶点对，则超图的Turán密度等于其拉格朗日密度。本项目将在申请人已有的工作基础上，主要研究几类3一致稀疏超图的拉格朗日密度及其扩张的Turán数。通过对目标超图进行结构分析，估算拉格朗日值，确定超图的拉格朗日密度，结合稳定性方法研究其扩张的Turán数和极值结构。本项目将通过对超图的拉格朗日密度的确定来进一步增进对超图Turán密度的了解。

Turán type problem is one of the central problems in extremal graph. The Lagrangian of a hypergraph has been a useful tool in Turán type problems. Recently, Lagrangian densities of hypergraphs and Turán numbers of their extensions have been studied actively. Sidorenko showed that the Lagrangian density of a hypergraph is the Turán density of its extension. Moreover, the Turán density of a hypergraph is the Lagrangian density if every pair of vertices of the hypergraph  is contained in an edge. The proposal will focus on the Lagrangian densities of hypergraphs and Turán numbers of their extensions. Combining hypergraph structural analysis,we determine the Lagrangian densities of hypergraphs. Using the stability method, we study the Turán number and the extremal structure of their extensions. One purpose of the proposal is to better understand Turán densities via Lagrangian densities of hypergraphs.