Small van der Waerden numbers

Determine W(r,k), the least N such that every r-colouring of {1,…,N} contains a monochromatic k-term arithmetic progression. Only seven non-trivial values are known; the open cases W(2,7), W(3,5), W(4,4) and W(5,3) invite better lower-bound colourings and exact computations.

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@misc{cairn-van-der-waerden-numbers,
  title        = {Small van der Waerden numbers},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/van-der-waerden-numbers}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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The problem

W(r,k) is the least N such that every colouring of {1, …, N} with r colours contains a monochromatic arithmetic progression of length k. Lower bounds come from explicit colourings; exact values need exhaustive (typically SAT-based) proofs.

Known status. The known non-trivial values are W(2,3) = 9, W(2,4) = 35, W(3,3) = 27 (Chvátal 1970), W(2,5) = 178 (Stevens–Shantaram 1978), W(4,3) = 76 (Beeler–O'Neil 1979), W(2,6) = 1132 (Kouril–Paul

and W(3,4) = 293 (Kouril 2012). Open cases with current lower bounds (from Wikipedia's table,

largely Rabung–Lotts cyclic "zipping" constructions and Monroe's distributed search over primes): W(2,7) > 3703, W(2,8) > 11495, W(3,5) > 2173, W(4,4) > 1048, W(5,3) > 170.

What counts as progress

A new lower bound: an explicit colouring of {1..N} beating a listed bound.
An exact value (the smallest open candidate is W(5,3)) with a checkable UNSAT proof.
Reproducible structured searches (power-residue/Rabung colourings, cyclic or palindromic colourings) with code and logs, including documented negative results such as "no palindromic colouring of length N exists".
Improved upper bounds for small open cases via SAT with symmetry breaking.

How it is checked — certificate format. A lower-bound certificate is a header "r k N" followed by a single string of N symbols from {0..r−1}, the colour of 1, 2, …, N. A short script checks every progression a, a+d, …, a+(k−1)d inside {1..N} (O(N²/k) progressions) and reports any monochromatic one; a colouring passing the check proves W(r,k) > N. An exact value ships the CNF generator plus an LRAT/DRAT proof that the length-W instance is unsatisfiable, re-checked with cake_lpr or drat-trim.

Current state

The problem