Erdős Problem #629

Partial progress by Gemini 3.1 Pro and GPT-5.4 Thinking (4 Apr 2026): partial result.

Problem
Erdős problem #629 (graph theory, chromatic number); current status: open
Credit
Gemini 3.1 Pro and GPT-5.4 Thinking · AI on its own; related literature found afterwards
Checked
Listed in Tao's wiki
Timeline
First AI contribution 2026-04-04 · wiki data as of 2026-06-30
Sources
Problem page (erdosproblems.com)Tao's AI-contributions wiki
PARTIALPer Tao's AI-contributions wiki

The problem

The list chromatic number χL(G)\chi_L(G) is defined to be the minimal kk such that for any assignment of a list of kk colours to each vertex of GG (perhaps different lists for different vertices) a colouring of each vertex by a colour on its list can be chosen such that adjacent vertices receive distinct colours.

Determine the minimal number of vertices n(k)n(k) of a bipartite graph GG such that χL(G)>k\chi_L(G)>k.

Statement quoted from erdosproblems.com/629, maintained by Thomas Bloom. Original source: [ERT80]. Current status there: open.

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 Literature
4 Apr, 2026 AI on its own; related literature found afterwards Gemini 3.1 Pro, GPT-5.4 Thinking 🟡 Partial result 🟡 Erdős (1964) + Erdős, Rubin, and Taylor (1980)

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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