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.