Section 49.4 Current Events and References
António Girão proved the Canonical Polynomial van der Waerden Theorem in 2020 in his paper titled A canonical polynomial van der Waerdan's theorem, arXiv:2005.04135.
Theorem 49.4.1. Canonical Polynomial van der Waerden Theorem.
Let \(\{p_1(x),\ldots, p_k(x)\}\) be a set of polynomials such that \(p_i(x) \in \mathbb{Z}[x]\) and \(p_i(0)=0\text{,}\) for every \(i \in \{1,\ldots, k\}\text{.}\) Then, in any colouring of \(\mathbb{Z}\text{,}\) there exist \(a, d \in \mathbb{Z}\) such that \(\{a+p_1(d), \ldots, a+p_k(d)\}\) forms either a monochromatic or a rainbow set.
Hillel Furstenberg. (2023, November 18). In Wikipedia. https://en.wikipedia.org/wiki/Hillel_Furstenberg.
M. K. Goh, 'Ergodic theory and arithmetic progressions', 2020,. https://marcelgoh.ca/misc/expo/furstenberg.pdf.
András Sárközy. (2023, November 18). In Wikipedia. https://en.wikipedia.org/wiki/András Sárközy.
Vitaly Bergelson. (2023, March 17). In Wikipedia. https://en.wikipedia.org/wiki/Vitaly_Bergelson.
Alexander Leibman. Ohio State University. https://math.osu.edu/people/leibman.1.
K. Dajani, S. Dirksin, A Simple Introduction to Ergodic Theory, 2008. https://webspace.science.uu.nl/~kraai101/lecturenotes2009.pdf.
Ergodic Theory, StudySmarter UK.https://www.studysmarter.co.uk/explanations/math/geometry/ergodic-theory/.
Mark Walters, Queen Mary University of London.https://webspace.maths.qmul.ac.uk/m.walters/.