Universal elements and the complexity of certain classes of infinite graphs

Abstract

A class of graphs has a universal element G0, if every other element of the class is isomorphic to an induced subgraph of G0. In Sections 1–4 we give a survey of some recent developments in the theory of universal graphs in the following areas: (1) Graphs universal for isometric embeddings, (2) universal random graphs, (3) universal graphs with forbidden subgraphs, (4) universal graphs with forbidden topological subgraphs. Section 5 is devoted to the problem of deciding how far a class of graphs G is from having a universal element. We introduce a new measure of the complexity of the class G, denoted by cp(G). This is defined to be the minimum cardinal κ such that there exist κ elements in G with the property that any other element of G can be embedded into at least one of them as an induced subgraph. G has a universal element if and only if cp(G)=1. Among other theorems we prove that (i) the complexity of the class of all countable graphs without n ⩾ 2 independent edges is finite; (ii) for any cardinal G, ω1⩽κ⩽2ω, it is consistent that the complexity of the class of all locally finite countable graphs is equal to κ. In Section 6 we consider some analogous questions for hypergraphs.