#### Alladi Ramakrishnan Hall

#### Topological and combinatorial van der Waerden theorem

#### Dibyendu De

##### University of Kalyani

*One of the old celebrated Ramsey theoretic result is van der Waerden*

theorem( vdW), which states that whenver the set of integers is partitioned

into finitely many colors, then there exists one monchromatic arbitrarily

long (but finite) arithmetic progressions.

Van der Waerden first proof of this theorem in 1927 using only combinatorial

methods. In this lecture we will prove vdW with methods from topological

dynamics. In fact we will provide a topological version of vdW (vdWt

for short) and will establish the equivalence of of vdW vdWt. The

proof is due to Furstenberg and Weiss \cite{key-2}.

Bergelson and Liebman proved a polynomial generalization of van der

Waerden's theorem (PvdW for short) \cite{key-1}. Again we provide

a topological version of PvdW (PvdWt for short) and will establish

the equivalence of of vdW vdWt.

Done