Almost-spanning universality in random graphs (Extended abstract)

A graph G is said to be H(n,Δ)-universal if it contains every graph on n vertices with maximum degree at most Δ. It is known that for any ε>0 and any natural number Δ there exists c>0 such that the random graph G(n,p) is asymptotically almost surely H((1−ε)n,Δ)-universal for p≥c(log⁡n/n)1/Δ. Bypassing this natural boundary, we show that for Δ≥3 the same conclusion holds when .