Symmetric monochromatic subsets in colorings of the Lobachevsky plane

TL;DR

This paper proves that in any finite Borel partition of the Lobachevsky plane, at least one cell contains an unbounded, centrally symmetric subset, revealing a fundamental symmetry property of such partitions.

Contribution

It establishes a new symmetry result for partitions of the Lobachevsky plane, extending combinatorial and geometric understanding of Borel colorings in hyperbolic geometry.

Findings

Existence of unbounded centrally symmetric subsets in at least one cell

Applicable to any finite Borel partition of the Lobachevsky plane

Advances understanding of symmetry in hyperbolic geometric partitions

Abstract

We prove that for each partition of the Lobachevsky plane into finitely many Borel pieces one of the cells of the partition contains an unbounded centrally symmetric subset.