Erdős Problem #1014

Full solution by OpenAI internal model (23 Apr 2026).

Problem
Erdős problem #1014 (graph theory, ramsey theory); current status: proved (Lean)
Credit
OpenAI internal model · AI on its own
Checked
Listed in Tao's wiki
Timeline
First AI contribution 2026-04-23 · 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 R(k,l)R(k,l) be the Ramsey number, so the minimal nn such that every graph on at least nn vertices contains either a KkK_k or an independent set on ll vertices.

Prove, for fixed k≥3k\geq 3, that

lim⁡l→∞R(k,l+1)R(k,l)=1.\lim_{l\to \infty}\frac{R(k,l+1)}{R(k,l)}=1.

Statement quoted from erdosproblems.com/1014, maintained by Thomas Bloom. Original source: [Er71,p.99]. Current status there: proved (Lean).

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
23 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.

∎