Erdős Problem #74: locally almost bipartite graphs of infinite chromatic number
resolvedconfidence 70%
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
Precise statement
Erdős, Hajnal and Szemerédi asked whether, for every function $f(n)\to\infty$, however slowly, there exists a graph of infinite chromatic number such that every finite $n$-vertex subgraph can be made bipartite by deleting at most $f(n)$ edges.
The answer is no: there exists a function $f(n)\to\infty$ such that every graph for which every finite $n$-vertex subgraph is within $f(n)$ edge deletions of bipartite has finite chromatic number.
The source statement is reproduced for indexing with attribution. Mathematical correctness requires domain-expert or mechanical review. VibeMath has not independently audited statement
fidelity, correctness, priority, or novelty.
What AI did
GPT-6 Astra (pre-release)
GPT-6 Astra autonomously found the disproof and wrote the Lean proofs in the FrontierMath Erdős benchmark, with no human seeing or steering the proof search. Six successful resolutions are included. They construct a sufficiently slowly diverging local bipartization budget and show that any graph satisfying it must actually be 3-colorable, contradicting the required infinite chromatic number. The proofs use several apparently distinct mechanisms, including finite profile exclusion, local defect witnesses, odd-cycle elimination with controlled gluing, and compactness.
Lean-verified. Checked here on 6 September 2026 from a clone of tadamcz/erdos74 at a626ecc: 11,481 lines of Lean, zero sorry outside the statement stubs, zero axiom declarations, no native_decide, unsafe or implemented_by; Comparator configuration present and CI runs it with only propext, Quot.sound and Classical.choice. The statement is copied verbatim from Formal Conjectures' ErdosProblems/74.lean at commit 488aade2 and the theorem is its negation. Six independent resolutions are included; all prove the stronger 3-colourability, while the compared theorem asserts only finite chromatic number. The rate f(n) ~ log n / log log n is not part of the certified theorem. erdosproblems.com, the field's own record, marks the problem DISPROVED (LEAN) with a proof exposition by Thomas Bloom, which is why this is Resolved rather than Candidate: the canonical tracker has accepted it.
Astra proves that there exists a divergent function
$$
f:\mathbb N\to\mathbb N,\qquad f(n)\to\infty,
$$
such that no graph $G$ of infinite chromatic number can satisfy
$$
d_{\mathrm{bip}}(H)\le f(n)
$$
for every finite $n$-vertex subgraph $H\subseteq G$, where $d_{\mathrm{bip}}(H)$ is the minimum number of edges that must be deleted to make $H$ bipartite.
In fact, every included resolution proves the stronger statement that graphs satisfying the constructed local bound are 3-colorable. The formal challenge advertises only the weaker conclusion that their chromatic number must be finite.
Known method families
argument (source-reported)
Source-reported tools: argument.
Independent: unknown · difference confidence: 0
What remains uncertain
VibeMath has not independently audited the mathematical statement, proof, or novelty claim.
VibeMath has not independently verified the mathematical claim.
AI-attempt independence and training-data exposure are unknown unless explicitly documented.
VibeMath has not independently audited the mathematical statement, proof, or novelty claim.