The tree embedding problem for digraphs

The tree embedding problem seeks to determine the minimal conditions a graph $G$ must satisfy to ensure it contains all trees with $k$ edges. A well-known conjecture by Erd\H{o}s and Sós states that any graph $G$ with $n$ vertices and more than $(k-1)n/2$ edges must contain every tree with $k$ edges. This conjecture was later extended by Addario-Berry et al. into the Antitree Conjecture, which asserts that any digraph $D$ with $n$ vertices and more than $(k-1)n$ arcs contains every antidirected tree with $k$ arcs.

In this talk, we present a proof of the Antitree Conjecture for the case where the digraph $D$ avoids certain orientations of the complete bipartite graph $K_{2,s}$, with $s = k/12$. Additionally, we discuss a proof of this conjecture for antidirected caterpillars. This is joint work with Maya Stein.