A Ramsey-type phenomenon in two and three dimensional simplices

TL;DR

This paper establishes a Ramsey-type theorem for subsets of 2D and 3D simplices, potentially leading to new proofs of extreme amenability in certain homeomorphism groups.

Contribution

It introduces a novel Ramsey-like theorem for low-dimensional simplices, linking combinatorial properties to topological group extreme amenability.

Findings

Developed a Ramsey-like theorem for 2D and 3D simplices

Potential to generalize to higher dimensions

Implications for proving extreme amenability of homeo groups

Abstract

We develop a Ramsey-like theorem for subsets of the two and three-dimensional simplex. A generalization of the combinatorial theorem presented here to all dimensions would produce a new proof that $\textrm{Homeo}_+[0,1]$ is extremely amenable (a theorem due to Pestov) using general results of Uspenskij on extreme amenability in homeomorphism groups.