Lagrangian densities of enlargements of matchings in hypergraphs

摘要

Determining the Turán density of a hypergraph is a central and challenging question in extremal combinatorics. We know very few about the Turán densities of hypergraphs. Even the conjecture given by Turán on the Turán density of the smallest complete hypergraph remains open. It turns out that the hypergraph Lagrangian method, a continuous optimization method has been helpful in hypergraph extremal problems. In this paper, we intend to explore this method further, and try to understand the Turán densities of hypergraphs via Lagrangian densities.Given an integer n and an r-uniform graph H, the Turán number of H, denoted by ex(n, H), is the maximum number of edges in an n-vertex r-uniform graph without containing a copy of H. The Turán density of H, denoted by π(H), is the limit of the function as n → ∞. The Lagrangian density of H is where λ(F) is the Lagrangian of F. For any r-uniform graph H, Sidorenko showed that πλ(H) equals the Turán density of the extension of H. So researching on Lagrangian densities of hypergraphs is helpful to better understand the behavior of the Turán densities of hypergraphs. For a t-vertex r-uniform graph H, since doesn’t contain H, where is the -vertex complete r-uniform graph. We say that H is λ-perfect if the equality holds, i.e., . A result given by Motzkin and Straus shows that every graph is λ-perfect. It is natural and fundamental to explore which hypergraphs are λ-perfect. Sidorenko (1989) showed that the -fold enlargement of a tree satisfying the Erdős-Sós conjecture with order at least Ar is λ-perfect, where Ar is the last maximum of the function as x ≥ 2. By using the so-called generalised Lagrangian of hypergraphs, Jenssen (2017) showed that the -fold enlargement of for or 7 is λ-perfect, where is the r-uniform matching of size s. The result given by Sidorenko (1989) implies that the -fold enlargement of for and t ≥ 4, or and t ≥ 6, or and t ≥ 8 is λ-perfect. So there are still some gaps between the results of Jenssen and Sidorenko. In this paper we fill the gaps for or 6. We also determine the Lagrangian densities for the -fold enlargement of for or 6.