An abstract, Hales-Jewett type extension of the polynomial van der Waerden
Theorem [J. Amer. Math. Soc. 9 (1996),725-753] is established:
Theorem. Let r,d,q \in \N. There exists N \in \N such that for any r-coloring
of the set of subsets of V={1,...,N}^{d} x {1,...,q} there exist a set a
\subset V and a nonempty set \gamma \subseteq {1,...,N} such that a \cap
(\gamma^{d} x {1,...,q}) = \emptyset, and the subsets a, a \cup (\gamma^{d} x
{1}), a \cup (\gamma^{d} x {2}), ..., a \cup (\gamma^{d} x {q}) are all of the
same color.
This ``polynomial'' Hales-Jewett theorem contains refinements of many
combinatorial facts as special cases. The proof is achieved by introducing and
developing the apparatus of set-polynomials (polynomials whose coefficients are
finite sets) and applying the methods of topological dynamics.Comment: 43 pages, published versio