GPT-6 Astra proves the Erdős–Sós conjecture (1962) with a short counting argument; mathematicians race to simplify and extend it
In Epoch AI's FrontierMath Erdős runs, a pre-release GPT-6 Astra autonomously proved the Erdős–Sós conjecture (Erdős problem #548): every graph with average degree greater than k−2 contains every tree on k vertices. The proof is Lean-verified. Its short, elementary argument counts vertex orderings. Within three weeks, leading combinatorialists published simplified versions, and other authors used Astra to extend the method to hypergraphs (Kalai's conjecture) and to digraphs.
Key facts
- Conjecture posed by Erdős and Sós around 1962–63; Chung's collection of Erdős's graph problems called it 'one of the most tantalizing problems in extremal graph theory'; erdosproblems.com listed a $100 prize
- Found in 1 of 3 FrontierMath Erdős attempts ($363, 20 hours of working time); Lean proof in tadamcz/erdos548 (created 3 Sep 2026), written autonomously by the model
- Idea (per Bloom's summary): count pairs (π, j) where π orders the vertices and v1–vj is an edge. This equals 2m(n−1)!, while for a fixed tree T an induction bounds it by C(T) + (k−2)·n!, and C(T) = 0 if G has no copy of T
- Human expositions and simplifications: Thomas Bloom (erdosproblems.com), Riordan & Scott (arXiv 2609.15893, which also find the extremal graphs), David Wood (arXiv 2609.17877), Bryce Frederickson (arXiv 2609.21159, random cyclic orderings), Jay Cummings (arXiv 2609.32011, visual exposition), and Ben Golub (erdosproblems.com, written with AI assistance)
- Riordan & Scott: 'In a startling development, the Erdős–Sós conjecture was recently proved in full by GPT-6 Astra … with a short and ingenious argument'
- Extensions found by Astra when prompted by Mubayi & Verstraëte: Kalai's conjecture for tight trees in hypergraphs (arXiv 2609.08012, r = 2 is Erdős–Sós) and an Erdős–Sós analogue for Eulerian digraphs (arXiv 2609.10987); both papers say the proofs were found by GPT-6 Astra
- Parallel human work: Reed & Stein proved the dense case (k ≥ γn) 'without any use of AI', with a version ready in early August (arXiv 2609.05417); Santos, Stein & Williams adapted the Astra argument to antidirected trees
Science result
- Field
- mathematics / extremal graph theory
- Problem
- Erdős–Sós conjecture (Erdős problem #548) (open since 1962)
- Result
- Full proof: every n-vertex graph with more than (k−2)n/2 edges contains every tree on k vertices.
- AI system
- GPT-6 Astra (pre-release)
- Human role
- Autonomous discovery and Lean proof in Epoch's benchmark harness; humans checked, simplified and generalised it afterwards
- Verification
- Formal proof in Lean 4; several independent human expositions; not yet peer-reviewed
- Status
- confirmed
- Why surprising
- A decades-old central conjecture of extremal graph theory had a proof short enough to explain in a page, and humans had missed it.
What happened
The proof came out of the same Epoch AI / Thomas Bloom benchmark runs that disproved Erdős problem #1. Once the Lean proof and Bloom's exposition were public, combinatorialists quickly posted cleaner human versions. They include Oliver Riordan and Alex Scott, who also determined the extremal graphs, and David Wood. Dhruv Mubayi and Jacques Verstraëte prompted Astra to extend the method to hypergraphs and digraphs and published the results with the proofs credited to the model.
Why it matters
Erdős–Sós is one of the best-known conjectures in extremal graph theory, and this is among the clearest cases of an AI finding a genuinely new, short idea that experts call "ingenious" and "surprising". The follow-up papers show the method being absorbed into the field within weeks.
Changelog
- 2026-09-30: created
Related events
- Pre-release GPT-6 Astra disproves Erdős's 'first serious problem' (1931, $500) and proves the rational-exponents conjecture, all Lean-verified, in Epoch's FrontierMath Erdős runs ★★★★★
- OpenAI releases GPT-6 Astra, its first GPT-6 model ★★★★★
- Summer 2026 flood: dozens of named conjectures settled on arXiv with disclosed AI help (July–September catalogue) ★★★★
Sources (10)
- codeGitHub: tadamcz/erdos548 (Lean proof found by GPT-6 Astra)
- paperarXiv 2609.25050: FrontierMath Erdős, Appendix B.4
- discussionerdosproblems.com #548
- paperarXiv 2609.15893: A short proof of the Erdős–Sós Conjecture (Riordan, Scott)
- paperarXiv 2609.17877: The Erdős–Sós Theorem (Wood)
- paperarXiv 2609.21159: Erdős–Sós via random cyclic orderings (Frederickson)
- paperarXiv 2609.32011: A visual exposition of the proof discovered by GPT-6 Astra (Cummings)
- paperarXiv 2609.08012: Kalai's Conjecture for Tight Trees (Mubayi, Verstraëte; proof by GPT-6 Astra)
- paperarXiv 2609.10987: Erdős–Sós for digraphs (Mubayi, Verstraëte; proof by GPT-6 Astra)
- paperarXiv 2609.05417: The Erdős–Sós conjecture in dense graphs (Reed, Stein; no AI)
id: 2026-09-03-erdos-sos-conjecture-proved-gpt-6-astra · updated 2026-09-30 · open in the interactive timeline