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.
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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.