For every integer t≥ 1, every oriented tree T with t edges, and every loopless digraph D on n vertices with no repeated arcs, if opposite arcs are permitted, D is Eulerian—that is, deg_D^+(v)=deg_D^-(v) for every vertex v—and |A(D)|>(t-1)n, then D contains an oriented copy of T. The bound (t-1)n is sharp.

Erdős–Sós for digraphs

The problem asks for a sharp threshold guaranteeing every oriented tree in an Eulerian digraph. The September 2026 preprint by Dhruv Mubayi and Jacques Verstraete claims the full directed analogue, including the directed-path case.

Known results

Addario-Berry, Havet, Linhares Sales, Reed, and Thomassé conjectured the threshold $(k-1)n$ for embedding every antidirected tree with $k$ arcs.
Graham, 1970, proved a linear bound for the antidirected-tree problem.
Burr, 1982, obtained the bound $4kn$ .
Recent work proves dense, approximate, minimum-semidegree, and girth-restricted variants, not the stated Eulerian theorem.

September 2026 claimed solution

Mubayi and Verstraete's unrefereed preprint states a sharp theorem for oriented trees in Eulerian digraphs and explicitly credits GPT-6 Astra. No retrieved source supplies independent verification, a referee report, or confirmation that the result matches the exact problem.

Current status (as of September 2026): A complete solution is claimed in an unrefereed preprint, but the theorem and its AI attribution remain unverified.

Erdős–Sós for digraphs

Primary source

Additional references

Progress summary

Solutions 0