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Erdős Problem #550

Erdős problem #550 · erdosproblems.com/550

Let m1≤⋯≤mkm_1\leq\cdots\leq m_k and nn be sufficiently large. If TT is a tree on nn vertices and GG is the complete multipartite graph with vertex class sizes m1,…,mkm_1,\ldots,m_k, prove that R(T,G)≤(χ(G)−1)(R(T,Km1,m2)−1)+m1R(T,G)\leq (\chi(G)-1)(R(T,K_{m_1,m_2})-1)+m_1.

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Graph Theory, Ramsey Theory
Posed by
Paul Erdős, Ralph Faudree, Cecil Rousseau, Richard Schelp
Year posed
1985
Years open
41y
Solved
2026-06-22
Model
ChatGPT
Vendor
OpenAI
Collaborators
Eric Li
Verification
Unreviewed
Publication
Preprint
Significance
10 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Claimed proved in a preprint of E. Li; erdosproblems.com still lists the problem open pending human review

What the AI did

The paper's disclosure states ChatGPT was used for ideation, formulation, proof exploration and refinement, narrowing the search space, programming and orchestration, with the author taking responsibility for the final contents; forum readers describe it as an affirmative paper almost purely by AI.

Verification

AI screenings on the forum initially flagged issues that turned out to be PDF-parsing artifacts; a re-run against the TeX source found no issues. No independent human review yet, and the site's owner is explicitly reserving judgement.

Sources

erdosproblems.com/550

Discussion