#### 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.

Done