Hypergraphs not containing a tight tree with a bounded trunk
Zolt\'an F\"uredi, Tao Jiang, Alexandr Kostochka, Dhruv Mubayi,, Jacques Verstra\"ete

TL;DR
This paper proves an asymptotic version of Kalai's conjecture for tight r-trees with bounded trunk size in hypergraphs, extending previous results and providing new bounds related to the shadow size.
Contribution
It establishes an asymptotic bound for hypergraphs avoiding tight r-trees with bounded trunk size, generalizing prior work and confirming the conjecture for small trees.
Findings
Proved an asymptotic version of Kalai's conjecture for all tight trees with bounded trunk size.
Provided a short proof of Kalai's conjecture for tight r-trees with up to four edges.
Connected the results to the intersection shadow theorem for 3-uniform hypergraphs.
Abstract
An -uniform hypergraph is a tight -tree if its edges can be ordered so that every edge contains a vertex that does not belong to any preceding edge and the set lies in some preceding edge. A conjecture of Kalai [Kalai], generalizing the Erd\H{o}s-S\'os Conjecture for trees, asserts that if is a tight -tree with edges and is an -vertex -uniform hypergraph containing no copy of then has at most edges. A trunk of a tight -tree is a tight subtree such that every edge of has vertices in some edge of and a vertex outside . For , the only nontrivial family of tight -trees for which this conjecture has been proved is the family of -trees with trunk size one in [FF] from 1987. Our main result is an asymptotic version of Kalai's conjecture for all tight trees of…
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