Improved Bound for Colorings Without Symmetrically Colored k-APs

Deng, Tidor and Zhao asked whether $[N]$ admits a coloring with $N^{o(1)}$ colors and no symmetrically coloured 4-term arithmetic progression, giving an $O(N^{\log_{22}3})$ coloring. The paper gives an $O_k(N^{4/k^2})$ coloring of $[N]$ avoiding symmetrically coloured $k$ -APs for every even $k\ge4$ , improving the exponent.

Result: Proved(see note)
Status: Partial result
AI contribution: AI co-developed
Method: Construction
Field: Additive Combinatorics, Ramsey Theory
Posed by: Deng, Tidor and Zhao
Year posed: —
Years open: —
Solved: 2026-07-22
Model: GPT-5
Vendor: OpenAI
Collaborators: Ruizhe Shi, Yiqi Dong
Verification: Unreviewed
Publication: Preprint
Significance: 10 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

Improves the exponent rather than answering the asked question: whether an $N^{o(1)}$ colouring exists remains open.

What the AI did

The acknowledgements state the key construction idea was suggested during an interaction with GPT-5.

Verification

No independent review. The model is credited with the key construction idea rather than the surrounding analysis. Preprint, not refereed.

Source

arXiv

Improved Bound for Colorings Without Symmetrically Colored k-APs

What was actually shown

What the AI did

Verification

Source

Discussion