The Order of Long Rainbow Arithmetic Progressions

Let $T_k$ be the least $t$ such that every equinumerous $t$-coloring of $[tn]$ contains a rainbow $k$-term arithmetic progression. Jungic, Licht, Mahdian, Nesetril and Radoicic conjectured $T_k = \Theta(k^2)$; Conlon, Fox and Sudakov proved $T_k = O(k^2 \log k)$. The matching lower bound $T_k = \Omega(k^2 \log k)$ holds, so $T_k = \Theta(k^2 \log k)$ and the conjectured order is wrong.

Result

Disproved(see note)

Status

Resolved

AI contribution

AI-assisted

Method

Construction

Field

Ramsey theory

Posed by

Veselin Jungic, Jacob Licht, Mohammad Mahdian, Jaroslav Nesetril, Rados Radoicic

Year posed

2003

Years open

23y

Solved

2026-07-16

Model

Codex (GPT-5.6), Claude Code (Fable 5)

Vendor

OpenAI / Anthropic

Collaborators

Jesse Geneson

Verification

Unreviewed

Publication

Preprint

Significance

20 / 100*

Disclosed cost

—

Wikipedia

No dedicated article