The Near-Quadratic Elekes-Ronyai Expander Conjecture
The near-quadratic Elekes-Ronyai expander conjecture over predicts that a nonspecial polynomial expands any finite set to near-quadratic size. False: a fixed nonspecial quadratic polynomial, together with arbitrarily large finite sets of real algebraic integers, has image with a fixed power saving from quadratic size.
- Result
- Disproved
- Status
- Resolved
- AI contribution
- AI-discovered
- Method
- Construction
- Field
- Additive combinatorics
- Posed by
- Gyorgy Elekes, Lajos Ronyai
- Year posed
- 2000
- Years open
- 26y
- Solved
- 2026-06-15
- Model
- ChatGPT 5.5 Pro, Rethlas
- Vendor
- OpenAI / Frenzy Math
- Collaborators
- Jihao Liu
- Verification
- Unreviewed
- Publication
- Preprint
- Significance
- 25 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What the AI did
The abstract states that the main result was obtained by generative AI, particularly ChatGPT 5.5 Pro and the Rethlas system. The proof builds on a recent OpenAI construction of an infinite tower, so this is an AI result standing on another AI result.
Verification
Single-author arXiv preprint; not yet peer-reviewed.
Source
arXiv:2606.16738 - A counterexample to the near-quadratic Elekes-Ronyai expander conjecture over R