Lagrangian densities of linear forests and Turán numbers of their extensions

The Lagrangian of a hypergraph has been a useful tool in hypergraph extremal problems. Recently, Lagrangian densities of hypergraphs and Turán numbers of their extensions have been studied actively. However, determining the Lagrangian density of a hypergraph is not an easy task even for a “simple” hypergraph. For example, to determine the Lagrangian density of $K_4^3$ is equivalent to determine the Turán density of $K_4^3$ (a long standing conjecture of Turán). Hefetz and Keevash studied the Lagrangian density of the 3‐uniform matching of size 2. Pikhurko determined the Lagrangian density of a 4‐uniform tight path of length 2 and this led to confirm the conjecture of Frankl and Füredi on the Turán number of the $r$‐uniform generalized triangle for the case $r=4$. It is natural and interesting to consider Lagrangian densities of other “basic” hypergraphs. In this paper, we determine the Lagrangian densities for a class of 3‐uniform linear forests. For positive integers $s$ and $t$, let $P_{s,t}$ be the disjoint union of a 3‐uniform linear path of length $s$ and $t$ pairwise disjoint edges. In this paper, we determine the Lagrangian densities of $P_{s,t}$ for any $t$ and $s=2$ or 3. Applying a modified version of Pikhurko's transference argument used by Brandt, Irwin, and Jiang, we obtain the Turán numbers of their extensions.