TY - JOUR
T1 - Hypergraphs not containing a tight tree with a bounded trunk II
T2 - 3-trees with a trunk of size 2
AU - Füredi, Zoltán
AU - Jiang, Tao
AU - Kostochka, Alexandr
AU - Mubayi, Dhruv
AU - Verstraëte, Jacques
N1 - Funding Information:
Research supported by grant K116769 from the National Research, Development and Innovation Office NKFIH, and by the Simons Foundation Collaboration grant #317487.Research partially supported by National Science Foundation award DMS-1400249.Research of this author is supported in part by National Science Foundation grant DMS-1600592 and by grants 18-01-00353A and 16-01-00499 of the Russian Foundation for Basic Research.Research partially supported by National Science Foundation award DMS-1300138.Research supported by National Science Foundation award DMS-1556524.This research was partly conducted during an American Institute of Mathematics Structured Quartet Research Ensembles workshop, and we gratefully acknowledge the support of AIM. We also thank the referees for helpful comments.
Publisher Copyright:
© 2019 Elsevier B.V.
PY - 2020/4/15
Y1 - 2020/4/15
N2 - A tight r-tree T is an r-uniform hypergraph that has an edge-ordering e1,e2,…,et such that for each i≥2, ei has a vertex vi that does not belong to any previous edge and ei−vi is contained in ej for some j′ of a tight r-tree T is a tight subtree T′ of T such that vertices in V(T)∖V(T′) are leaves in T. Kalai's Conjecture was proved (Frankl and Füredi, 1987) for tight r-trees that have a trunk of size one. In a previous paper (Füredi et al., 2019) we proved an asymptotic version for all tight r-trees that have a trunk of bounded size. In this paper we continue that work to establish the exact form of Kalai's Conjecture for all tight 3-trees of at least 8 edges that have a trunk of size two.
AB - A tight r-tree T is an r-uniform hypergraph that has an edge-ordering e1,e2,…,et such that for each i≥2, ei has a vertex vi that does not belong to any previous edge and ei−vi is contained in ej for some j′ of a tight r-tree T is a tight subtree T′ of T such that vertices in V(T)∖V(T′) are leaves in T. Kalai's Conjecture was proved (Frankl and Füredi, 1987) for tight r-trees that have a trunk of size one. In a previous paper (Füredi et al., 2019) we proved an asymptotic version for all tight r-trees that have a trunk of bounded size. In this paper we continue that work to establish the exact form of Kalai's Conjecture for all tight 3-trees of at least 8 edges that have a trunk of size two.
KW - Extremal hypergraph theory
KW - Hypergraph trees
KW - Turán problem
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U2 - 10.1016/j.dam.2019.09.010
DO - 10.1016/j.dam.2019.09.010
M3 - Article
AN - SCOPUS:85073822662
SN - 0166-218X
VL - 276
SP - 50
EP - 59
JO - Discrete Applied Mathematics
JF - Discrete Applied Mathematics
ER -