# Axiomatic Theory of Betweenness

## TL;DR

This paper introduces a generalized concept of betweenness as a relation between sets of points and develops a logical system that characterizes its universal properties within graph theory.

## Contribution

It extends traditional betweenness to sets of points and provides a sound, complete logical framework for understanding this relation in graphs.

## Key findings

- Established a formal logical system for set-based betweenness.
- Proved the system's soundness and completeness.
- Unified geometric and graph-theoretic betweenness concepts.

## Abstract

Betweenness as a relation between three individual points has been widely studied in geometry and axiomatized by several authors in different contexts. The article proposes a more general notion of betweenness as a relation between three sets of points. The main technical result is a sound and complete logical system describing universal properties of this relation between sets of vertices of a graph.