GPT-6 Astra resolves Erdős Problem 74 on local bipartiteness

There is a divergent integer-valued function f such that if every finite n-vertex subgraph of a graph becomes bipartite after at most f(n) edge deletions, then the graph has finite chromatic number. The formal proof gives three-colourability.

Sources for this record

What was already known

Erdős, Hajnal and Szemerédi suspected infinitely chromatic examples for every arbitrarily slowly divergent f.

AI involvement AI-autonomous

GPT-6 Astra found the proof and wrote Lean code in a network-isolated run without human mathematical steering. Adamczewski packaged the repository; Bloom later explained its approach.

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

Importance Notable

The result settles the local-versus-global chromatic question in the opposite direction.

Limitations

The compared theorem does not provide an explicit rate for f. Bloom sketches the core argument but does not document a line-by-line proof check.

Technical verification Formal (Lean 4 with Mathlib and Comparator)

Source-reported Lean build and Comparator match to the trusted challenge in a pristine environment; no broader informal theorem is silently inferred.

Formal checking

compiled proof, trusted statement comparison, and standard-axiom check

Checker
Lean 4 kernel and FrontierMath Erdős Comparator
Scope
Problem #74 compared theorem
Basis
Reported by the linked checker
Evidence source

Checking a formal statement does not establish that it matches the original problem.

Human understanding Partial

Bloom describes the core induction, while the formal-to-informal details and claimed explicit rate remain undigested.

Correction and publication history

2026-10-18 · publication

The result settles the local-versus-global chromatic question in the opposite direction.

Read this published revision