Superexponential van der Waerden numbers

OpenAI's theorem (2026) that van der Waerden numbers grow super-exponentially for every number of colours, W_r(k) > k^(ck floor(log2 r)); through circle coordinates and adaptive meshes, a balanced colouring chosen by the local lemma, and a sparse perturbation by squared-norm bands.

Five lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.

1. Van der Waerden numbers
2. Arrangements, tails and the local lemma
3. Coordinates, meshes and label counts
4. A balanced colouring and the dichotomy
5. Norm bands and the two-colouring

Theorems and assumptions · Reader and editable sources · Course record

Prerequisites

• Elementary probability (independence, expectation, Markov's inequality), modular arithmetic and real analysis on the circle.