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 runiform graph H, the Turan number of H, denoted by ex(n, H), is the maximum number of edges in an nvertex runiform graph without containing a copy of H. The Turan density of H, denoted by xcfx80(H), is the limit of the function      e  x  (  n  ,  H  )    (   n  r   )      as nxc2xa0xe2x86x92xc2xa0xe2x88x9e. The Lagrangian density of H is      xcfx80  xcexbb    (  H  )     sup     r  !  xcexbb   (  F  )   :  H    i  s    n  o  t    c  o  n  t  a  i  n  e  d    i  n    F     ,     where xcexbb(F) is the Lagrangian of F. For any runiform graph H, Sidorenko showed that xcfx80xcexbb(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 tvertex runiform graph H,      xcfx80  xcexbb    (  H  )   xe2x89xa5  r  !  xcexbb   (   K   t  xe2x88x92  1   r   )      since     K   t  xe2x88x92  1   r     doesnxe2x80x99t contain H, where     K   t  xe2x88x92  1   r     is the     (  t  xe2x88x92  1  )    vertex complete runiform graph. We say that H is xcexbbperfect if the equality holds, i.e.,      xcfx80  xcexbb    (  H  )     r  !  xcexbb   (   K   t  xe2x88x92  1   r   )     . A result given by Motzkin and Straus shows that every graph is xcexbbperfect. It is natural and fundamental to explore which hypergraphs are xcexbbperfect. Sidorenko (1989) showed that the     (  r  xe2x88x92  2  )    fold enlargement of a tree satisfying the Erdxc5x91sSos conjecture with order at least Ar is xcexbbperfect, where Ar is the last maximum of the function      g  r    (  x  )      (  r    x  xe2x88x92  3  )       xe2x88x92  r     xe2x88x8f   i    1    r  xe2x88x92  1     (  i    x  xe2x88x92  2  )      as xxc2xa0xe2x89xa5xc2xa02. By using the socalled generalised Lagrangian of hypergraphs, Jenssen (2017) showed that the     (  r  xe2x88x92  2  )    fold enlargement of     M  2  2     for     r    5  ,  6     or 7 is xcexbbperfect, where     M  s  r     is the runiform matching of size s. The result given by Sidorenko (1989) implies that the     (  r  xe2x88x92  2  )    fold enlargement of     M  t  2     for     r    5     and txc2xa0xe2x89xa5xc2xa04, or     r    6     and txc2xa0xe2x89xa5xc2xa06, or     r    7     and txc2xa0xe2x89xa5xc2xa08 is xcexbbperfect. 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  xe2x88x92  3  )    fold enlargement of     M  t  3     for     r    5     or 6.

Extremal Problems for tPartite and tColorable Hypergraphs Fix integers t ge r ge 2 and an runiform hypergraph F. We prove that the maximum number of edges in a tpartite runiform hypergraph on n vertices that contains no copy of F is c_t, Fn choose r o(nr), where c_t, F can be determined by a finite computation.   We explicitly define a sequence F_1, F_2, ldots of runiform hypergraphs, and prove that the maximum number of edges in a tchromatic runiform hypergraph on n vertices containing no copy of F_i is alpha_t,r,in choose r o(nr), where alpha_t,r,i can be determined by a finite computation for each ige 1. In several cases, alpha_t,r,i is irrational. The main tool used in the proofs is the Lagrangian of a hypergraph.