The Maximum Cliques Of Non-uniform Hypergraphs And Polynomial Optimization Problems

Determining the Turán densities of hypergraphs has been one of the most challenging problems in extremal hypergraph theory.In 1965,Motzkin and Straus established a remarkable connection between the clique number and the Lagrangian of a graph,and gave another proof of the classical result of Turán on the Turán density of a complete graph by applying the connection.This aroused the great interests on exploring the method of continuous optimization on extremal problems.In 2009,Rota Bul?o and Pelillo generalized the Motzkin-Straus Theorem to r-uniform hypergraphs in [28] by establishing a one-to-one correspondence between the optimization problem of homogeneous polynomial function and the maximum clique of an r-uniform hypergraph,and obtained an upper bound of Turán densities of r-uniform complete hypergraphs.In this dissertation,we obtain a connection between the clique number and the minimum of a continuous optimization problem for a {1,r}-hypergraph,a {p,q,r,s}-hypergraph and a {p,q,r,s,t}-hypergraph.Applying the results,we obtain an upper bound for the Turán density of a complete {1,r}-hypergraph,{p,q,r,s}-hypergraph and {p,q,r,s,t}-hypergraph.