Erdős Problem #1091

Full solution by OpenAI internal model (9 Apr 2026).

Problem
Erdős problem #1091 (chromatic number, graph theory); current status: solved
Credit
OpenAI internal model · AI on its own
Checked
Listed in Tao's wiki
Timeline
First AI contribution 2026-04-09 · wiki data as of 2026-06-30
Sources
Problem page (erdosproblems.com)Tao's AI-contributions wiki
SOLVEDPer Tao's AI-contributions wiki

The problem

Let GG be a K4K_4-free graph with chromatic number 44. Must GG contain an odd cycle with at least two diagonals?

More generally, is there some f(r)→∞f(r)\to \infty such that every graph with chromatic number 44, in which every subgraph on ≤r\leq r vertices has chromatic number ≤3\leq 3, contains an odd cycle with at least f(r)f(r) diagonals?

Statement quoted from erdosproblems.com/1091, maintained by Thomas Bloom. Original source: [Er76c,p.4]. Current status there: solved.

AI contributions

As recorded in Terence Tao's wiki AI contributions to Erdős problems (data as of 30 Jun 2026). 🟢 full, 🟡 partial, ⚪ unverified, 🔴 incorrect.

Date Kind AI systems Outcome
9 Apr, 2026 AI on its own OpenAI internal model 🟢 Full solution

Caveats

The wiki's classifications are provisional and are not a benchmark. Many Erdős problems have never had a thorough literature review, so a result that looks new may later turn out to be known; some problems were also stated ambiguously or incorrectly. See the problem page for the full discussion.

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