Which subsets of an infinite random graph look random?

TL;DR

This paper investigates the properties of subsets of infinite random graphs, showing that sets of positive density are universal and those with divergent reciprocal sums are weakly universal, with sharpness and limitations explored.

Contribution

It establishes almost sure universality properties of subsets in infinite random graphs and characterizes the sharpness and limitations of these properties.

Findings

Sets of positive upper density are universal in almost every infinite random graph.

Sets with divergent reciprocal sums are weakly universal in almost every infinite random graph.

The results are sharp and do not extend to all partition regular families.

Abstract

Given a countable graph, we say a set $A$ of its vertices is \emph{universal} if it contains every countable graph as an induced subgraph, and $A$ is \emph{weakly universal} if it contains every finite graph as an induced subgraph. We show that, for almost every graph on $\mathbb N$, $(1)$ every set of positive upper density is universal, and $(2)$ every set with divergent reciprocal sums is weakly universal. We show that the second result is sharp (i.e., a random graph on $\mathbb N$ will almost surely contain non-universal sets with divergent reciprocal sums) and, more generally, that neither of these two results holds for a large class of partition regular families.