Suborbits in Knaster's problem

TL;DR

This paper explores connections between Euclidean Ramsey problems and Knaster-type problems, demonstrating weak Knaster properties for certain geometric configurations in spheres and Euclidean spaces using ideas from Ramsey theory.

Contribution

It introduces a novel link between Ramsey theory and geometric problems, proving weak Knaster properties for specific simplices and triangles.

Findings

Weak Knaster properties for non-equatorial triangles in spheres

Weak Knaster properties for simplices in Euclidean spaces

Establishment of a similarity between Euclidean Ramsey and Knaster problems

Abstract

In this paper we exhibit a similarity between Euclidean Ramsey problems and Knaster-type problems. By borrowing ideas from Ramsey theory we prove weak Knaster properties of non-equatorial triangles in spheres, and of simplices in Euclidean spaces.