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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.  conjecture of Kalai [Kalai], generalizing the ErdH{o}s-Su0027os 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 $Tu0027$ of a tight $r$-tree $T$ is a tight subtree such that every edge of $T-Tu0027$ has $r-1$ vertices in some edge of $Tu0027$ and a vertex outside $Tu0027$. For $rge 3$, 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 Kalaiu0027s 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 Kalaiu0027s 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].

Hypergraphs Not Containing a Tight Tree with a Bounded Trunk