  The theory of dense graph limits comes with a natural sampling process which
yields an inhomogeneous variant G(n,W) of the Erdos-Renyi random graph. Here we
study the clique number of these random graphs. We establish the concentration
of the clique number of G(n,W) for each fixed n, and give examples of graphons
for which G(n,W) exhibits wild long-term behavior. Our main result is an
asymptotic formula which gives the almost sure clique number of these random
graphs. We obtain a similar result for the bipartite version of the problem. We
also make an observation that might be of independent interest: Every graphon
avoiding a fixed graph is countably-partite.
