Student Number Theory Seminar

Shouzhen Gu, Department of Pure Mathematics, University of Waterloo

"The Hales-Jewett Theorem"

We give a proof of Schur's Theorem, which states that if the integers from 1 to $N$ are coloured using $r$ colours, then for sufficiently large $N$, there will be a monochromatic solution to $x+y=z$. Then, we present the Hales-Jewett Theorem on the existence of monochromatic combinatorial lines in the integer points of a hypercube of high enough dimension. As a corollary, we can find monochromatic arithmetic progressions of any given length in the integers from 1 to $N$ for large enough $N$, which demonstrates van der Waerden's Theorem.

MC 5403