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For L_p-norms (1 < p < ∞) in the plane, a two-coloring exists preventing monochromatic copies of any infinite set M. However, for polygonal norms, some infinite sets M guarantee monochromatic copies under any two-coloring.

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Area of Science:

  • Geometric measure theory
  • Combinatorial geometry
  • Functional analysis

Background:

  • The study of coloring geometric objects is a fundamental problem in mathematics.
  • Investigating the existence of monochromatic configurations under different distance metrics is crucial.
  • Understanding the properties of L_p-norms and polygonal norms is essential in geometric analysis.

Purpose of the Study:

  • To determine conditions under which a two-coloring of the plane can avoid monochromatic isometric copies of a given infinite set.
  • To explore the contrast between L_p-norms and polygonal norms regarding monochromatic configurations.
  • To establish the existence of sets that are guaranteed to form monochromatic copies under specific norm conditions.

Main Methods:

  • Utilizing concepts from Ramsey theory and topological methods.
  • Constructing specific two-colorings of the Euclidean plane.
  • Analyzing the properties of isometric embeddings in different metric spaces.

Main Results:

  • Proved that for any L_p-norm (1 < p < ∞) and any infinite set M in R^2, a two-coloring exists such that no isometric copy of M is monochromatic.
  • Demonstrated that for any polygonal norm in the plane, there exists an infinite set M such that every two-coloring contains a monochromatic isometric copy of M.

Conclusions:

  • The nature of the norm significantly influences the existence of monochromatic configurations.
  • L_p-norms offer greater flexibility in avoiding monochromatic sets compared to polygonal norms.
  • This research highlights the interplay between metric properties and combinatorial coloring principles.