# Completing graphs to metric spaces

## TL;DR

This paper investigates classes of metrically homogeneous graphs that exclude certain triangle configurations, proving they have the extension property for partial automorphisms and describing their Ramsey expansions.

## Contribution

It establishes the extension property for specific classes of triangle-avoiding metrically homogeneous graphs and characterizes their Ramsey expansions.

## Key findings

- Certain classes of metrically homogeneous graphs have the extension property for partial automorphisms.
- The paper describes Ramsey expansions for these classes.
- It identifies conditions under which these graphs omit specific triangle types.

## Abstract

We prove that certain classes of metrically homogeneous graphs omitting triangles of odd short perimeter as well as triangles of long perimeter have the extension property for partial automorphisms and we describe their Ramsey expansions.