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 πλ(F)=sup{r!λ(G):GisF-free}, 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.