These graphs are named after C. Ward Henson, who published a construction for them (for all i ≥ 3) in 1971.
(The existence of such a sequence uniquely defines the Rado graph.)
He then defines Gi 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.
[1] With this construction, each graph Gi is an induced subgraph of Gi + 1, and the union of this chain of induced subgraphs is the Rado graph itself.
Any finite or countable i-clique-free graph H can be found as an induced subgraph of Gi by building it one vertex at a time, at each step adding a vertex whose earlier neighbors in Gi match the set of earlier neighbors of the corresponding vertex in H. That is, Gi is a universal graph for the family of i-clique-free graphs.