The connection between polynomial optimization, maximum cliques and Turán densities

Abstract In 1965, Motzkin–Straus established the connection between the maximum cliques and the Lagrangian of a graph, the maximum value of a quadratic function determined by a graph in the standard simplex. This connection gave a proof of the Turan’s classical result on Turan densities of complete graphs. In 1980’s, Sidorenko and Frankl–Furedi further developed this method for hypergraph Turan problems. However, the connection between the Lagrangian and the maximum cliques of a graph cannot be extended to hypergraphs. In 2009, S. Rota Bulo and M. Pelillo defined a homogeneous polynomial function of degree r determined by an r -uniform hypergraph and gave the connection between the minimum value of this polynomial function and the maximum cliques of an r -uniform hypergraph. In this paper, we provide a connection between the local (global) minimizers of non-homogeneous polynomial functions to the maximal (maximum) cliques of hypergraphs whose edges containing r − 1 and r vertices. This connection can be applied to obtain an upper bound on the Turan densities of complete { r − 1 , r } -type hypergraphs.