An absolute constant C suffices for bounded-degree tree universality in G(n, C ln n/n), and cycle-factor universality is optimal up to constants via depth-optimal linking systems.
Embedding bounded degree spanning trees in random graphs
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We prove that if a tree $T$ has $n$ vertices and maximum degree at most $\Delta$, then a copy of $T$ can almost surely be found in the random graph $\mathcal{G}(n,\Delta\log^5 n/n)$.
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Universality in random graphs via optimal linking systems: trees and beyond
An absolute constant C suffices for bounded-degree tree universality in G(n, C ln n/n), and cycle-factor universality is optimal up to constants via depth-optimal linking systems.