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Lagrangian densities of linear forests and Turán numbers of their extensions
Author(s) -
Hu Sinan,
Peng Yuejian,
Wu Biao
Publication year - 2020
Publication title -
journal of combinatorial designs
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.618
H-Index - 34
eISSN - 1520-6610
pISSN - 1063-8539
DOI - 10.1002/jcd.21687
Subject(s) - mathematics , hypergraph , lagrangian , disjoint sets , conjecture , combinatorics , pairwise comparison , path (computing) , discrete mathematics , statistics , computer science , programming language
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 , tbe 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 , tfor 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.