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, $G_{3}$ 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, $G_{3}$, is also called the homogeneous triangle-free graph or the universal triangle-free graph.

Construction
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 $G_{i}$.

Universality
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.

Because there exist $i$-clique-free graphs of arbitrarily large chromatic number, the Henson graphs have infinite chromatic number. More strongly, if a Henson graph $G_{i}$ is partitioned into any finite number of induced subgraphs, then at least one of these subgraphs includes all $i$-clique-free finite graphs as induced subgraphs.

Symmetry
Like the Rado graph, $G_{3}$ contains a bidirectional Hamiltonian path such that any symmetry of the path is a symmetry of the whole graph. However, this is not true for $G_{i}$ when $i > 3$: for these graphs, every automorphism of the graph has more than one orbit.