Evolutions of finite graphs

TL;DR

This paper explores the simultaneous construction of countable graphs from finite graphs through embeddings and projections, analyzing properties of universal and profinite graphs, including the Rado graph and Henson's triangle-free graph.

Contribution

It introduces a combined approach to constructing countable graphs from finite graphs using both embeddings and projections, and studies the properties of the resulting universal and profinite graphs.

Findings

The Rado graph can be constructed using both embeddings and projections.

Henson's universal triangle-free graph cannot be constructed this way.

The projectively universal profinite graph has a dense subset of isolated vertices.

Abstract

Every countable graph can be built from finite graphs by a suitable infinite process, either adding new vertices randomly or imposing some rules on the new edges. On the other hand, a profinite topological graph is built as the inverse limit of finite graphs with graph epimorphisms. We propose to look at both constructions simultaneously. We consider countable graphs that can be built from finite ones by using both embeddings and projections, possibly adding a single vertex at each step. We show that the Rado graph can be built this way, while Henson's universal triangle-free graph cannot. We also study the corresponding profinite graphs. Finally, we present a concrete model of the projectively universal profinite graph, the projective Fraisse limit of finite graphs, showing in particular that it has a dense subset of isolated vertices.