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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 for this record

What was already known

The conjecture had many special cases, but no published proof covering every tree at the sharp density. Disjoint cliques show the bound is tight.

AI involvement AI-autonomous

GPT-6 Astra found the permutation-counting argument and wrote the Lean proof without human guidance during its benchmark run. Bloom, Wood, Riordan and Scott later explained or simplified the mathematics.

Benchmark and repository document no human guidance during the model proof search.

Importance Major

Resolves a longstanding extremal graph problem.

Limitations

The public Lean challenge assumes one extra edge in an odd-parity case. The stronger classical threshold is supported by a separate counting lemma and later human proofs; those proofs do not audit the original Lean code.

Technical verification Formal (Lean 4 with Mathlib and Comparator)

Lean/Comparator checks the benchmark theorem with its parity caveat; the stronger classical statement is supported by a separate counting lemma and later human proofs.

Independent checking

compiled proof and challenge comparison

Checker
Lean 4 kernel and FrontierMath Erdős Comparator
Scope
Erdos548.erdos_548 benchmark theorem
Basis
Reported by the linked checker
Evidence source

Human understanding Full

Bloom, Wood and Riordan–Scott explain and simplify the permutation-counting mechanism.

Correction and publication history