Max-norm Ramsey theory

Given a metric space M that contains at least two points, the chromatic number chi (Rno, M) is defined as the minimum number of colours needed to colour all points of an n -dimensional space Rno with the max -norm such that no isometric copy of M is monochromatic. The last two authors have recently shown that the value chi (Rno, M) grows exponentially for all finite M. In the present paper we refine this result by giving the exact value chi M such that chi (Rno, M) = (chi M + o(1))n for all 'one-dimensional' M and for some of their Cartesian products. We also study this question for infinite M. In particular, we construct an infinite M such that the chromatic number chi (Rno, M) tends to infinity as n -> o. (c) 2024 Elsevier Ltd. All rights reserved.