For any r, k there exists W(r, k) such that every r-colouring of {1, …, W(r, k)} contains a monochromatic arithmetic progression of length k. Small values: W(2, 3) = 9, W(2, 4) = 35, W(2, 5) = 178. Gowers 2001 tower bound.
For any r, k there exists W(r, k) such that every r-colouring of {1, …, W(r, k)} contains a monochromatic arithmetic progression of length k. Small values: W(2, 3) = 9, W(2, 4) = 35, W(2, 5) = 178. Gowers 2001 tower bound.