  Let $G$ be a graph and $S, T \subseteq V(G)$ be (possibly overlapping) sets
of terminals, $|S|=|T|=k$. We are interested in computing a vertex sparsifier
for terminal cuts in $G$, i.e., a graph $H$ on a smallest possible number of
vertices, where $S \cup T \subseteq V(H)$ and such that for every $A \subseteq
S$ and $B \subseteq T$ the size of a minimum $(A,B)$-vertex cut is the same in
$G$ as in $H$. We assume that our graphs are unweighted and that terminals may
be part of the min-cut. In previous work, Kratsch and Wahlstr\"om (FOCS
2012/JACM 2020) used connections to matroid theory to show that a vertex
sparsifier $H$ with $O(k^3)$ vertices can be computed in randomized polynomial
time, even for arbitrary digraphs $G$. However, since then, no improvements on
the size $O(k^3)$ have been shown.
  In this paper, we draw inspiration from the renowned Bollob\'as's
Two-Families Theorem in extremal combinatorics and introduce the use of total
orderings into Kratsch and Wahlstr\"om's methods. This new perspective allows
us to construct a sparsifier $H$ of $\Theta(k^2)$ vertices for the case that
$G$ is a DAG. We also show how to compute $H$ in time near-linear in the size
of $G$, improving on the previous $O(n^{\omega+1})$. Furthermore, $H$ recovers
the closest min-cut in $G$ for every partition $(A,B)$, which was not
previously known. Finally, we show that a sparsifier of size $\Omega(k^2)$ is
required, both for DAGs and for undirected edge cuts.
