Let \(1\leq \alpha \leq \beta \leq \gamma\) be cardinals, and let \({\mathcal G}_{\gamma}(K_{\alpha,\beta})\) denote the class of all graphs on \(\gamma\) vertices having no subgraphs isomorphic to \(K_{\alpha,\beta}\). A graph \(G_ 0\in {\mathcal G}_{\gamma}(K_{\alpha,\beta})\) is called universal if every \(G\in {\mathcal G}_{\gamma}(K_{\alpha,\beta})\) can be embedded into \(G_ 0\) as a subgraph. We prove that, if \(\alpha <\omega \leq \gamma\) and the general continuum hypothesis is assumed, then \({\mathcal G}_{\gamma}(K_{\alpha,\beta})\) has a universal element if and only if (i) \(\gamma >\omega\) or (ii) \(\gamma =\omega\), \(\alpha =1\) and \(\beta\leq 3\). Using the axiom of constructibility, we also show that there does not exist a universal graph in \({\mathcal G}_{\omega_ 1}(K_{\omega,\omega_ 1})\).