Lagrangian densities of some sparse hypergraphs and Turán numbers of their extensions.

The Lagrangian of a hypergraph has been a useful tool in hypergraph extremal problems. The Lagrangian density of an r -uniform graph F is π λ ( F ) = sup { r ! λ ( G ) : G i s F - f r e e } , where λ ( G ) is the Lagrangian of an r -uniform graph G . Recently, Lagrangian densities of hypergraphs and Turán numbers of their extensions have been studied actively. In particular, Hefetz and Keevash (2013) studied the Lagrangian density of the 3-uniform matching of size 2 and the Turán number of its extension. We obtain the Lagrangian densities of a 3-uniform matching of size t , a 3-uniform linear star of size t , and a 4-uniform linear star of size t . Using a stability argument of Pikhurko and a transference technique between the Lagrangian density of an r -uniform graph F and the Turán number of its extension, we can also determine the Turán numbers of their extensions. [ABSTRACT FROM AUTHOR]