On universal graphs without cliques or without large bipartite graphs

Abstract: "For every uncountable cardinal [lambda], suitable negations of the Generalized Continuum Hypothesis imply: (a) There is no universal K[subscript ╬▒╬▓]-free graphs in [lambda] for all infinite ╬▒ and ╬▓. (b) There is no universal K[subscript ╬▒]-free graph in [lambda] for all ╬▒ [> or =] 3. The instance K[subscript omega,omegaΓéü] for [lambda] = NΓéü was settled in [KP] from a strengthening of CH."