Andrey Kupavskii: Max-norm Ramsey theory

By analogy with Euclidean Ramsey theory, we ask the following general question: given a metric space X, what is the smallest number of colors needed to color R^d_{infty} (i.e., the d-dimensional real affine space with maximum norm) so that the coloring does not contain a monochromatic isometric copy of X? We show that for any finite metric space X this number is exponential in d. We also relate it to several intriguing finite combinatorial problems and show that, to a large extent, this question is 1-dimensional. Joint work with Nóra Frankl and Arsenii Sagdeev.