Lagrangian densities of linear forests and Turán numbers of their extensions

Abstract

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 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.

The Lagrangian density of{;123,234,456} and the Tur\'an number of its extension

“…Recall that T P 3 is the 3-graph with vertex set [6] and edge set {123, 234, 456}. In this section, we will show that the maximum possible Lagrangian among all T P 3 -free 3-graphs is uniquely achieved by K 3 5 .…”

“…In this paper, we obtain the Lagrangian density of a 3-uniform path of length 3 where two consecutive edges have 1 or 2 vertices in common. Precisely, let T P 3 be the 3-graph with vertex set [6] and edge set {123, 234, 456}. We show that the Lagrangian density of T P 3 achieves only on K 3 5 .…”

“…Jiang-Peng-Wu in [7] obtained the Lagrangian density of a 3-uniform matching of any size and the Turán number of the extension. The authors of [19] and [6] determined the Lagrangian density of a 3-uniform linear path of length 3 or 4, the Lagrangian density of the disjoint union of a 3-uniform linear path of length 2 or 3 and a matching of any size, and the corresponding Turán numbers of their extensions. It seems to be more difficult if we replace a linear path by a tight path.…”

An irrational Lagrangian density of a single hypergraph

“…Sidorenko [31] applied Lagrangians of hypergraphs to first find infinitely many Turán densities of hypergraphs. More recent developments of this method were obtained in [26,2,13,24,3,25,15,16,14,37]. Determining the Lagrangian of a hypergraph is much more difficult than graphs and there is no conclusion similar to Theorem 1.2 for hypergraphs.…”

Lagrangian densities of short 3-uniform linear paths and Turán numbers of their extensions

“…An r-graph is linear if any two edges have at most 1 vertex in common. Hu, Peng and Wu [8], and Chen, Liang and Peng [3] showed that the disjoint union of a 3-uniform linear path of length 2 or 3 and a 3-uniform matching, and the disjoint union of a 3-uniform tight path of length 2 and a 3-uniform matching are perfect. Yan and Peng [24] showed that the 3-uniform linear cycle of length 3 ({123, 345, 561}) is perfect, and F 5 ({123, 124, 345}) is not perfect (by determining its Lagrangian density).…”