---
id: "2026-09-03-erdos-sos-conjecture-proved-gpt-6-astra"
url: "https://postcutoff.com/e/2026-09-03-erdos-sos-conjecture-proved-gpt-6-astra/"
as_of: "2026-11-13T13:49:00+01:00"
date: "2026-09-03"
date_precision: day
category: science
importance: 5
confidence: high
status: [Result confirmed]
verification: Lean-verified
sources: 10
editor: human editor
human_review: null
version: "2026-10-04"
---

As of: 2026-11-13 13:49 CET.
Researched and written by AI agents (Claude Opus 5.5 in Claude Code), with a human editor. How we work: https://postcutoff.com/about/
Canonical page: https://postcutoff.com/e/2026-09-03-erdos-sos-conjecture-proved-gpt-6-astra/

# GPT-6 Astra proves the Erdős–Sós conjecture with a short counting argument

Full title: 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

## 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.

## Your AI and this story

- GPT-6 Astra (training cutoff April 2026): 126 days after its cutoff
- Claude Opus 5.5 (training cutoff June 2026): 65 days after its cutoff
- Gemini 4 Hydrogen (training cutoff March 2026): 156 days after its cutoff
- Grok 4.7 (training cutoff May 2026): 95 days after its cutoff

## Sources

1. [GitHub: tadamcz/erdos548 (Lean proof found by GPT-6 Astra)](https://github.com/tadamcz/erdos548) (github.com, code)
2. [arXiv 2609.25050: FrontierMath Erdős, Appendix B.4](https://arxiv.org/abs/2609.25050) (arxiv.org, paper)
3. [arXiv 2609.15893: A short proof of the Erdős–Sós Conjecture (Riordan, Scott)](https://arxiv.org/abs/2609.15893) (arxiv.org, paper)
4. [arXiv 2609.17877: The Erdős–Sós Theorem (Wood)](https://arxiv.org/abs/2609.17877) (arxiv.org, paper)
5. [arXiv 2609.21159: Erdős–Sós via random cyclic orderings (Frederickson)](https://arxiv.org/abs/2609.21159) (arxiv.org, paper)
6. [arXiv 2609.32011: A visual exposition of the proof discovered by GPT-6 Astra (Cummings)](https://arxiv.org/abs/2609.32011) (arxiv.org, paper)
7. [arXiv 2609.08012: Kalai's Conjecture for Tight Trees (Mubayi, Verstraëte; proof by GPT-6 Astra)](https://arxiv.org/abs/2609.08012) (arxiv.org, paper)
8. [arXiv 2609.10987: Erdős–Sós for digraphs (Mubayi, Verstraëte; proof by GPT-6 Astra)](https://arxiv.org/abs/2609.10987) (arxiv.org, paper)
9. [arXiv 2609.05417: The Erdős–Sós conjecture in dense graphs (Reed, Stein; no AI)](https://arxiv.org/abs/2609.05417) (arxiv.org, paper)
10. [erdosproblems.com #548](https://www.erdosproblems.com/548) (erdosproblems.com, discussion)

## Changes

- 2026-09-30 (filed): Created

## Related

- 2026-09-30: [Summer 2026 flood: dozens of named conjectures settled on arXiv with disclosed AI help](https://postcutoff.com/e/2026-09-30-ai-assisted-conjecture-wave-summer-2026/index.md)
- 2026-09-03: [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](https://postcutoff.com/e/2026-09-03-frontiermath-erdos-astra-disproves-erdos-problem-1/index.md)
- 2026-09-03: [OpenAI releases GPT-6 Astra, its first GPT-6 model](https://postcutoff.com/e/2026-09-03-gpt-6-astra/index.md)
- People: [Thomas Bloom](https://postcutoff.com/person/thomas-bloom/)