In the Erdös-Rényi model for random graphs,the clique number is seen to go to infinity as the number of vertices grows. Is anyone aware of models for random graphs with bounded clique number?



Model for random graphs where clique number remains bounded
The Erdös-Rényi model works. One just has to take the associated probability $p$ to scale with the size of the graph $n$. For instance, Theorem 4.13 in *Random Graphs* by Bollobás shows that for ER graph with $p=p(n)$ such that
$np→∞$ with $np=o(n^\frac{1}{3})$ as $n→∞$, the clique number satisfies $\omega(G(n, p))= 3$ with high probability. 


