Lagrangian densities of enlargements of matchings in hypergraphs

Abstract Determining the Turan density of a hypergraph is a central and challenging question in extremal combinatorics. We know very few about the Turan densities of hypergraphs. Even the conjecture given by Turan on the Turan 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 Turan densities of hypergraphs via Lagrangian densities. Given an integer n and an r-uniform graph H, the Turan 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 Turan 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 πλ(H) equals the Turan density of the extension of H. So researching on Lagrangian densities of hypergraphs is helpful to better understand the behavior of the Turan 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-Sos conjecture with order at least Ar is λ-perfect, where Ar 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.