Henson graph

In graph theory, the Henson graph G_i is an undirected infinite graph, the unique countable homogeneous graph that does not contain an i-vertex clique but that does contain all K_i-free finite graphs as induced subgraphs. For instance, G₃ is a triangle-free graph that contains all finite triangle-free graphs.

These graphs are named after C. Ward Henson, who published a construction for them (for all i ≥ 3) in 1971. The first of these graphs, G₃, is also called the homogeneous triangle-free graph or the universal triangle-free graph.

To construct these graphs, Henson orders the vertices of the Rado graph into a sequence with the property that, for every finite set S of vertices, there are infinitely many vertices having S as their set of earlier neighbors. (The existence of such a sequence uniquely defines the Rado graph.) He then defines G_i to be the induced subgraph of the Rado graph formed by removing the final vertex (in the sequence ordering) of every i-clique of the Rado graph.

With this construction, each graph G_i is an induced subgraph of G_{i + 1}, and the union of this chain of induced subgraphs is the Rado graph itself. Because each graph G_i omits at least one vertex from each i-clique of the Rado graph, there can be no i-clique in G_i.

Any finite or countable i-clique-free graph H can be found as an induced subgraph of G_i by building it one vertex at a time, at each step adding a vertex whose earlier neighbors in G_i match the set of earlier neighbors of the corresponding vertex in H. That is, G_i is a universal graph for the family of i-clique-free graphs.

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