Infinite closed monochromatic subsets of a metric space

TL;DR

This paper proves that in any coloring of k-element subsets of an uncountable separable metric space, there exists an infinite monochromatic subset containing its limit point, extending Ramsey theory to metric spaces.

Contribution

It introduces a new result in metric space Ramsey theory, demonstrating the existence of infinite monochromatic subsets with limit points under coloring conditions.

Findings

Existence of infinite monochromatic subsets with limit points in uncountable separable metric spaces.

Extension of classical Ramsey theory to metric space contexts.

Provides a new combinatorial property for colored metric spaces.

Abstract

Given a coloring of the k-element subsets of an uncountable separable metric space, we show that there exists an infinite monochromatic subset which contains its limit point.