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Breakthrough Tracker record

A human exposition and Lean artifact support a claimed full proof of the Erdős–Sós conjecture

Riordan and Scott claim that every graph of average degree greater than k−2 contains every k-vertex tree, including a classification of equality cases and an antidirected-tree analogue. Their mathematical exposition builds on a GPT-6 Astra proof, and a separate Lean 4 artifact checks the formal statement.

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Stable ID
math-erdos-sos-full-proof-formalization-2026
Revision
math-erdos-sos-full-proof-formalization-2026.v2
Field
Mathematics · Extremal graph theory and formal verification
Evidence
Tier 1 · Peer reviewed: No
Record state
Provisional · Formalized provisional proof
Last checked

AI role

The underlying full proof is attributed to GPT-6 Astra. Riordan and Scott supplied a conventional human-authored exposition; a separate Lean 4 repository provides machine checking of the encoded statement.

Record details

Problem or result
The full Erdős–Sós conjecture for embedding every k-vertex tree
Authors
Oliver Riordan and Alex Scott; underlying proof credited to GPT-6 Astra; Lean artifact maintained by Tamás C. Adamczewski
Institutions
University of Oxford
Result date
Preprint first submitted September 14; v3 posted October 8, 2026

Why it matters

This is the full 1960s conjecture rather than the dense-regime theorem already tracked. The short human exposition and separate formal artifact materially improve checkability.

Limits

The v3 manuscript remains unrefereed. Its authors describe minor corrections in Section 5 and add an Eulerian-graph result; these changes do not independently validate the claimed full proof or alter the formal artifact's semantic-review caveat.

Sources

  1. Primary: Riordan and Scott, full Erdős–Sós conjecture
  2. Proof artifact: GPT-6 Astra proof artifact for Erdős problem 548
  3. Formal proof: Lean 4 repository for Erdős problem 548
  4. Expert exposition: Thomas Bloom, proof exposition for Erdős problem 548
  5. Revised primary: Erdős–Sós full-proof preprint v3

Correction and revision history

  1. 2026-10-02 — Added after primary-source, scope, status, AI-role and limitation review.
  2. 2026-10-19 — Recorded arXiv v3's minor Section 5 corrections and added Eulerian-graph result; retained formalized provisional-proof status.

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