# Local Finiteness of Infinite Neighbor Complexes

## TL;DR

This paper proves that for infinite subsets of ^n, if their neighbor complex has finite dimension, then each vertex in the complex has finitely many neighbors, linking geometric properties to local finiteness.

## Contribution

It establishes a new connection between the finite-dimensionality of neighbor complexes and local finiteness of vertices for infinite subsets of ^n.

## Key findings

- Finite-dimensional neighbor complexes imply finite neighbors per vertex.
- The result applies to infinite subsets of ^n.
- Provides a criterion for local finiteness based on complex dimension.

## Abstract

We show that if the neighbor complex, as defined by H.~Scarf, of an infinite subset of $\Z^n$ has finite dimension, then each vertex has finitely many neighbors.