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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 K 4 3 , 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 e x (n , H) (n r) as n → ∞. The Lagrangian density of H is π λ (H) = sup { r ! λ (F) : H i s n o t c o n t a i n e d i n F } , where λ (F) is the Lagrangian of F. For any r -uniform graph H , Sidorenko showed that π λ (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 , π λ (H) ≥ r ! λ (K t − 1 r) since K t − 1 r doesn't contain H , where K t − 1 r is the (t − 1) -vertex complete r -uniform graph. We say that H is λ -perfect if the equality holds, i.e., π λ (H) = r ! λ (K t − 1 r). 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 (r − 2) -fold enlargement of a tree satisfying the Erdős-Sós conjecture with order at least A r is λ -perfect, where A r is the last maximum of the function g r (x) = (r + x − 3) − r ∏ i = 1 r − 1 (i + x − 2) as x ≥ 2. By using the so-called generalised Lagrangian of hypergraphs, Jenssen (2017) showed that the (r − 2) -fold enlargement of M 2 2 for r = 5 , 6 or 7 is λ -perfect, where M s r is the r -uniform matching of size s. The result given by Sidorenko (1989) implies that the (r − 2) -fold enlargement of M t 2 for r = 5 and t ≥ 4, or r = 6 and t ≥ 6, or r = 7 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 r = 5 or 6. We also determine the Lagrangian densities for the (r − 3) -fold enlargement of M t 3 for r = 5 or 6. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]