Abstract:
We'll generalize last week's talk from colorings of \([n]\) to colorings of \([m]^n\) and show there is always an \(n\) large enough to find a monochromatic 'combinatorial line.' This is theorem is due to Hales and Jewett. The theorem from Van Der Waerden from last week is a fairly straightforward corollary of the Hales-Jewett theorem as we'll also see. If time permits we'll talk a bit about Rado's theorem and what it means for a matrix with rational entries to be partition regular.