Max-norm Ramsey theory

Frankl, Nóra and Kupavskii, A. and Sagdeev, Arsenii (2024) Max-norm Ramsey theory. EUROPEAN JOURNAL OF COMBINATORICS, 118. ISSN 0195-6698

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Abstract

Given a metric space M that contains at least two points, the chromatic number χ (Rn ∞, M) is defined as the minimum number of colours needed to colour all points of an n-dimensional space Rn ∞ with the max-norm such that no isometric copy of M is monochromatic. The last two authors have recently shown that the value χ (Rn ∞, M) grows exponentially for all finite M. In the present paper we refine this result by giving the exact value χM such that χ (Rn ∞, M) = (χM + 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 χ (Rn ∞, M) tends to infinity as n → ∞.

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