Antidirected trees in digraphs

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

A finite simple digraph has no loops and at most one arc for each ordered pair of distinct vertices; opposite arcs may both occur. An antidirected tree is an orientation of a tree in which each vertex has indegree zero or outdegree zero. Its order is its number of vertices. Containment means an injective map preserving arcs, not necessarily an induced copy.

Statement recorded in the supplied source.

For every integer $k\ge2$, does every digraph $D$ with \[|A(D)|>(k-2)|V(D)|\] contain every antidirected tree of order $k$? The same source records the undirected Erdős--Sós special case: does every simple graph $G$ with $|E(G)|>(k-2)|V(G)|/2$ contain every tree of order $k$? [S2]

Short English statement

Does the sharp arc-density condition force every antidirected tree, including the corresponding undirected tree conjecture?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3261 (OPG-46279). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open Problem Garden, Antidirected trees in digraphs, original node 46279, statement and mathematical discussion. https://www.openproblemgarden.org/op/antidirected_trees_in_digraphs.