Lagrangian densities of linear forests and Turán numbers of their extensions

Abstract

AbstractThe 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 is equivalent to determine the Turán density of (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 ‐uniform generalized triangle for the case . 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 and , let be the disjoint union of a 3‐uniform linear path of length and pairwise disjoint edges. In this paper, we determine the Lagrangian densities of for any and 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.