Chapter 49 Polynomial van der Waerden Theorem
Project by: Mark Collins,, Damanjit Heer, and Jacky Lim.
\(\textbf{Summary:}\) Our project focused on understanding and mastering the polynomial van der Waerden theorem and its combinatorial proof presented in a paper by Mark Walters. Our goal was to understand the proof and put together a presentation that outlines the proof in detail for people to analyze and understand. However, for our presentation in class our goal was to simplify the proof such that not all the details were presented but the main ideas were covered. This was done to meet the time constraint and allow students to follow along. We studied the history of the proof, first that Hillel Furstenberg and András Sárközy independently proved the case with a single polynomial. Then we looked at Vitaly Bergelson and Alexander Leibman as they were able to prove the polynomial van der Waerdan Theorem in 1996 using ergodic theory. Then, the one we focused on was the combinatorial proof using methods of double induction and colour focusing presented by Mark Walters.