# Small subgraphs and their extensions in a random distance graph

## TL;DR

This paper extends previous results on threshold probabilities for random distance graphs to arbitrary graphs, showing that the count of strictly balanced subgraphs follows an asymptotic Poisson distribution at the threshold.

## Contribution

It generalizes earlier findings to all graphs and establishes the Poisson distribution of subgraph counts at the threshold.

## Key findings

- Threshold probabilities for arbitrary graphs are established.
- Number of strictly balanced subgraphs follows a Poisson distribution asymptotically.
- Results extend previous work on specific graph classes.

## Abstract

In previous papers, threshold probabilities for the properties of a random distance graph to contain strictly balanced graphs were found. We extend this result to arbitrary graphs and prove that the number of copies of a strictly balanced graph has asymptotically Poisson distribution at the threshold.