HJ numbers revisited

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, $|\Lambda|$) 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.