The Erdős–Sós tree conjecture
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The Erdős–Sós tree conjecture asks whether every finite simple graph of average degree greater than k−1 contains every tree on k+1 vertices. The strict edge threshold matters when (k−1)n is odd.
Why it matters and what counts as progress
Why it matters: The average-degree threshold would force every tree of a given size, a general extremal embedding principle.
What counts as progress: Prove the full strict-threshold embedding theorem or obtain stronger bounds for restricted tree families.
How the problem developed
- 1963 · Conjecture [1]
Paul Erdős and Vera T. Sós Posed the conjecture that every n-vertex graph with more than (k−2)n/2 edges contains every tree on k vertices.
- 15 Sept 2026 · Proof [2]
GPT-6 Astra, with the proof written up by Oliver Riordan and Alex Scott Reported a proof of the full tree-embedding conjecture; the authors’ note gives a simplified argument and characterizes extremal graphs.
Bryce Frederickson Presented a simplified proof using random cyclic orderings and gave an additional result for antidirected trees in digraphs.
AI contribution
AI-found proof of the Erdős–Sós tree conjecture
For n≥k+1, every simple graph with at least (k−1)n/2+1 edges contains every tree on k+1 vertices. A separate counting lemma also reaches the classical strict threshold when (k−1)n is odd.
Sources
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