Universal Graphs without Large Cliques

The theory of universal graphs originated from the observation of R. Rado [4,5] that a universal countable graph X exists, i.e., X is countable and isomorphically embeds every countable graph. He also showed that under GCH, there is a universal graph in every infinite cardinal. Since then, several results have been proved about the existence of universal elements in different classes of graphs. For example, a construction similar to Rado’s shows, that for every natural number n ≥ 3, there is a universalK(n)-free countable graph, or, if GCH is assumed, there is one in every infinite cardinal (here K(n) denotes the complete graph on n vertices). This result also follows from the existence theorem of universal and special models. The following folklore observation shows that this cannot be extended to K(ω). Assume that X = (V,E) is a K(ω)-free graph of cardinal λ that embeds every K(ω)-free graph of cardinal λ. Let a 6∈ V , and define the graph X ′ on V ′ = V ∪ {a} as follows. X ′ on V is identical with X, a is joined to every vertex of V . Clearly, X ′ is K(ω)-free. So, by assumption, there is an embedding g:V ′ → V of X ′ into X. Put a0 = a, and, by induction, an+1 = g(an). As g is edge preserving, we get, by induction on n, that an is joined to every at with t > n, so they are distinct, and form a K(ω) in X , a contradiction. In Section 1 we give some existence/nonexistence statements on universal graphs, which under GCH give a necessary and sufficient condition for the existence of a universal graph of size λ with no K(κ), namely, if either κ is finite or cf(κ) > cf(λ). The special case when λ = λ was first proved by F. Galvin. In Section 2 we investigate the question that if there is no universal K(κ)-free graph of size λ then how many of these graphs embed all the other. It was proved in [1], that if λ = λ (e.g., if λ is regular and the GCH holds below λ), and κ = ω, then this number is λ. We show that this holds for every κ ≤ λ of countable cofinality. On the other hand, even for κ = ω1, and any regular λ ≥ ω1 it is consistent that the GCH holds below λ, 2 λ is as large as we wish, and the above number is either λ or 2, so both extremes can actually occur. Similar results when the excluded graphs are disconnected, were proved in [2] and [3].