>k-homogeneous infinite graphs

TL;DR

This paper classifies countably infinite graphs that are k-homogeneous for some k, revealing the precise conditions under which such graphs are non-homogeneous, based on Ramsey theory insights.

Contribution

It provides an explicit classification of countably infinite k-homogeneous graphs, detailing when they are non-homogeneous based on their 1- and 2-homogeneity.

Findings

Non-homogeneous k-homogeneous graphs are characterized by failures in 1- or 2-homogeneity.

Classification relies on Ramsey theory to distinguish between homogeneous and non-homogeneous cases.

The paper offers a complete description of the structure of these graphs.

Abstract

In this article we give an explicit classification for the countably infinite graphs $\mathcal{G}$ which are, for some $k$, $\geq$$ k$-homogeneous. It turns out that a $\geq$$k-$homogeneous graph $\mathcal{M}$ is non-homogeneous if and only if it is either not $1-$homogeneous or not $2-$homogeneous, both cases which may be classified using ramsey theory.