  A tree-based network $\mathcal N$ on $X$ is universal if every rooted binary
phylogenetic $X$-tree is a base tree for $\mathcal N$. Hayamizu and,
independently, Zhang constructively showed that, for all positive integers $n$,
there exists an universal tree-based network on $n$ leaves. For all $n$,
Hayamizu's construction contains $\Theta(n!)$ reticulations, while Zhang's
construction contains $\Theta(n^2)$ reticulations. A simple counting argument
shows that an universal tree-based network has $\Omega(n\log n)$ reticulations.
With this in mind, Hayamizu as well as Steel posed the problem of determining
whether or not such networks exists with $O(n\log n)$ reticulations. In this
paper, we show that, for all $n$, there exists an universal tree-based network
on $n$ leaves with $O(n\log n)$ reticulations.
