The Lagrangian Density Of 3 Uniform Hypergraph And The Turán Number Of Its Extention

Turán problems are central to combinatorics.Given a positive integer n and an r-uniform hypergraph F,the Turán number ex(n,F)is the maximum number of edges in an F-free r-uniform hypergraph on n vertices.The Turán density of F is defined asWe know quite well about the Turán number of a nonbipartite graph asymptot-ically:For a complete graph,its Turán number was given by Turán in 1941 and the uniqueness of its extremal graph was confirmed.Forageneralgraph,Erd6s-Stone?Simonovits associated the Turán number with its chromatic number.However,for a bipartite graph,we know quite few about its Turán number.For example,known results of even length cycles merely have:C4,C6,C10.For hypergraphs,there are very few known results on Turán numbers.For exam-ple,determining the Turán number of the simplest complete graph K43 is not completely solved yet.With the development of Turán problems,a lot of useful tools have been formed.Lagrangian is one of the most important tools in the study of Turán problems.Hefetz-Keevash points out that it is interesting to explore how large the La-grangian of hypergraphs with some properties can be.In this paper,given a hypergraph F,we focus on the maximum of Lagrangians among hypergraphs without F as a sub-graph,which we call Lagrangian density.In recent years,several researchers have been working on the Lagrangian density and the Turán number of hypergraph extension.For example,in the paper "A hypergraph Turán theorem via lagrangians of intersecting families,J.Combin.Theory Ser.A(2013),2020-2038",Hefetz-Keevash determined the Lagrangian density of a 3-uniform 2 matching,i.e.,{123,456},and applied it to obtain the Turán number of its extension.Let TPa be a 3 uniform hypergraph whose vertex set is[6],and edge set is{123,234,456}.In this thesis,we obtain that the Lagrangian density of TPa achieved on K53,complete the 3-uniform hypergraph with 5 vertices.Applying it,we get that the Turán number of its extension is ex(n,H6TP3)=t53(n),where T53(n)is the balanced complete 5 partite 3 uniform graph on n vertices,and t53(n)is the edge number of T53(n).We also show that the unique extremal hypergraph is the balanced complete 5 partite 3 uniform hypergraph on n vertices.