These are the slides from a talk I gave on Wednesday 11th March 2020, to the VUW Logic Seminar. The topic was the Ramsey-theoretic problem of partition regularity of Diophantine equations, and in particular, the work I did with Lupini and Moreira on this problem over the summer of 2018/19.
Abstract: An old question in Ramsey theory asks whether, given a Diophantine equation, we can find a monochromatic solution in any finite colouring of the natural numbers. Rado completely solved the linear case in 1933, but the nonlinear case has proved much harder. In recent years, nonstandard analysis, as well as ergodic theory, have been applied fruitfully by Di Nasso and others to yield new results in this area. I will give an overview of recent breakthroughs obtained by myself, Lupini and Moreira by applying nonstandard methods to the partition regularity problem.
PR-of-DE_v2_handout