Leírás
An $r$-uniform hypergraph is a {\it 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, 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 {\em 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 $T'$. Our main result is an asymptotic version of Kalai's conjecture for all tight trees $T$ of bounded trunk size. This follows from a Katona type upper bound on the size of the shadow of a $T$-free hypergraph.
Joint work with Jiang, Kostochka, Mubayi and Verstraete