Hypergraphs not containing a tight tree with a bounded trunk

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 $r$-uniform hypergraph is a tight $r$-tree if its edges can be ordered so that every edge $e$ contains a vertex $v$ that does not belong to any preceding edge and the set $e-v$ lies in some preceding edge. A conjecture of Kalai [Kalai], generalizing the Erd\H{o}s-S\'os Conjecture for trees, asserts that if $T$ is a tight $r$-tree with $t$ edges and $G$ is an $n$-vertex $r$-uniform hypergraph containing no copy of $T$ then $G$ has at most $\frac{t-1}{r}\binom{n}{r-1}$ edges. A trunk $T'$ of a tight $r$-tree $T$ is a tight subtree such that every edge of $T-T'$ has $r-1$ vertices in some edge of $T'$ and a vertex outside $T'$. For $r\ge 3$, the only nontrivial family of tight $r$-trees for which this conjecture has been proved is the family of $r$-trees with trunk size one in [FF] from 1987. Our main result is an asymptotic version of Kalai's conjecture for all tight trees $T$ of bounded trunk size. This follows from our upper bound on the size of a $T$-free $r$-uniform hypergraph $G$ in terms of the size of its shadow. We also give a short proof of Kalai's conjecture for tight $r$-trees with at most four edges. In particular, for $3$-uniform hypergraphs, our result on the tight path of length $4$ implies the intersection shadow theorem of Katona [Katona].