Universal graphs without large cliques

Abstract

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 lambda with no K(kappa), namely, if either kappa is finite or cf(kappa)>cf(lambda). (Here K(kappa) denotes the complete graph on kappa vertices.) The special case when lambda^{< kappa}= lambda was first proved by F. Galvin. Next, we investigate the question that if there is no universal K(kappa)-free graph of size lambda then how many of these graphs embed all the other. It was known, that if lambda^{< lambda}= lambda (e.g., if lambda is regular and the GCH holds below lambda), and kappa = omega, then this number is lambda^+. We show that this holds for every kappa <= lambda of countable cofinality. On the other hand, even for kappa = omega_1, and any regular lambda >= omega_1 it is consistent that the GCH holds below lambda, 2^{lambda} is as large as we wish, and the above number is either lambda^+ or 2^{lambda}, so both extremes can actually occur.