Maximisers of the hypergraph Lagrangian outside the principal range

摘要

The Lagrangian of a hypergraph is a function that has featured notably in hypergraph Turán densities. Motzkin and Straus established the relationship between Lagrangian and the maximum clique in a graph. As a generalization of Motzkin-Straus Theorem, Frankl and Füredi put forward a well-known conjecture, which states that the r-graph with m edges formed by taking the first m sets in the colex ordering of N(r) has the largest Lagrangian of all r-graphs with m edges. The conjecture was settled when r=3 for sufficiently large m, and is false outside the principal range for r≥4, proved by Gruslys, Letzter and Morrison. Until now, it is still open to characterize the maximisers of the hypergraph Lagrangian outside the principal range for r≥4. In this paper, we study the conjecture outside the principal range. We determine the smallest number of edges for which the conjecture is false, and also partially characterize the maximisers of the Lagrangian.