Hypergraphs not containing a tight tree with a bounded trunk ~II: 3-trees with a trunk of size 2
Zolt\'an F\"uredi, Tao Jiang, Alexandr Kostochka, Dhruv Mubayi and, Jacques Verstra\"ete

TL;DR
This paper advances the understanding of Kalai's conjecture by proving the exact threshold for the containment of certain tight 3-trees with a trunk of size two in large hypergraphs, extending previous asymptotic results.
Contribution
It establishes the exact edge threshold for embedding all tight 3-trees with a trunk of size two and at least 20 edges in large hypergraphs, confirming Kalai's conjecture for this class.
Findings
Proved the exact threshold for tight 3-trees with a trunk of size two.
Extended previous asymptotic results to exact thresholds.
Confirmed Kalai's conjecture for a new class of hypergraphs.
Abstract
A tight -tree is an -uniform hypergraph that has an edge-ordering such that for each , has a vertex that does not belong to any previous edge and is contained in for some . Kalai conjectured in 1984 that every -vertex -uniform hypergraph with more than edges contains every tight -tree with edges. A trunk of a tight -tree is a tight subtree of such that vertices in are leaves in . Kalai's Conjecture was proved in 1987 for tight -trees that have a trunk of size one. In a previous paper we proved an asymptotic version of Kalai's Conjecture for all tight -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 -trees with at least…
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