2026/07/16 by Saharon Shelah
#math.CO
We improve the bounds on the Hales-Jewett numbers to a tower of exponentiations. Earlier it was WaW (that is, iterations of towers which are themselves iterated exponentiations). We improve the inductive step there (induction on the size of the alphabet, |Λ|) to 2-exponentiations, instead of towers. In the longer work in typing, (A) We present this inductive step as a partition theorem in its own right; (but in this preliminary version we make it just serve the bound on HJ numbers). (B) We shall deal with the density version of Hales-Jewett with similar bound. We are also dealing with the Graham-Rothschild Theorem and the Affine Ramsey Theorem and the polynomial case, and give background.