On universal graphs without cliques or withour large bipartite graphs

Abstract

For every uncountable cardinal $λ$, suitable negations of the Generalized Continuum Hypothesis imply: - For all infinite $α$ and $β$, there is no universal $K_{α,β}$-free graphs in $λ$ - For all $α\ge 3$, there is no universal $K_α$-free graph in $λ$ The instance $K_{ω,ω_1}$ for $λ=\aleph_1$ was settled by Komjath and Pach from the principle $\diamondsuit(ω_1)$.