The Maximum Lagrangian of 5-uniform Hypergraphs without Containing Two Edges Intersecting at a Vertex

Obtaining Turán densities of hypergraph through determining Lagrangian densities have been studied actively.Sidorenko showed that the Lagrangian density of an r-uniform hypergraph equals the Turán density of the extension of this hypergraph.Indeed,Lagrangian density and Turán density are the same for a large class of hypergraphs such as hypergraphs covering pairs.Let Pr,sbe the r-uniform hypergraph with two edges intersecting at s vertices.Sidorenko determined the Lagrangian densities of Pr,sfor r=3 and s=1 or 2,or r=4 and s=2 or 3.Hefetz and Keevash determined for r=3 and s=0,and Bene Watts,Norin and Yepremyan determined for r≥4 and s=0.The cases r=5 and s=4 or r=6 and s=5 were determined by Norin and Yepremyan.Jenssen showed for r=5,6 or 7,and s=3,4 or 5 respectively.The cases r=4 and s=1 or r=5 and s=2 were determined by Jiang,Peng and Wu,and Hu,Peng and Wu respectively.For r=5,the only remaining case is s=1.In this paper,we determine for this case.