Alladi Ramakrishnan Hall

Topological and combinatorial Hales-Jewett Theorem

Dibyendu De

University of Kalyani

In the previous lecture we have discussed about vdW and vdWt. In 1963
Hales and Jewett established a far reaching generalization of vdW
called Hales-Jewett Theorem (HJ for short). There are various form
of HJ. We will present here the form in set theoretic language. Like
vdWt, we will show that HJ also an equivalent topological version
(HJt for short).

Being a general combinatorial fact about sets, the Hales- Jewett Theorem
does not appeal to any special properties of the set of natural numbers,
besides its cardinality. But we will show that how vdW follows from
HJ.

In this lecture we obtain the polynomial Hales-Jewett Theorem (PHJ),
which extends the Polynomial van der Waerden Theorem above in the
way similar to that in which the Hales-Jewett Theorem generalizes
the classical van der Waerden Theorem. We will also provide topological
version of PHJ named as PHJt, which will be proved to be equivalent
with PHJ. The idea of the proof goes back to the seminal paper of
Furstenberg and Weiss \cite{key-3}.

The equivalence of Theorems PHJ and PHJt suggests that in dealing
with Ramsey-theoretical questions one has at his disposal not only
the conventional language of combinatorics but also an equivalent
language or, rather, method of topological dynamics.

To derive the combinatorial proposition from a topological one, one
considers the (compact) space of all r-colorings of the set in question
and applies to it a general topological recurrence theorem. On the
other hand, to derive a topological recurrence theorem from a combinatorial
statement about monochromatic configurations which one always finds
in any finite coloring, one uses the basic property of compact spaces,
namely the existence of finite covers consisting of sets of arbitrarily
small diameter, and, after inducing a finite coloring on the acting
(semi) group with the help of such a cover, utilizes the combinatorial
statement. We will allow ourselves to pass freely from the combinatorial
language to the topological one and back when describing and deriving
examples and corollaries of PHJ.