Let $G$ be a bipartite graph with bipartition $A,B$ where $A$ consists of all 2-subsets of $[n]$, $B$ consists of all 3-subsets of $[n]$ and the edges are defined by the inclusion relation. I would like to know the smallest number of vertices in $A$ that cover all of $B$. In other words: what is the size of the smallest subset $S$ of $A$ such that every vertex of $B$ has an edge to $S$.
The first values seem to be $1,2,4,6,9,12,16$ (starting with $n=3$), but Sloane gives too many hits for this pattern.
I know that the general problem is hard, but this specific one is highly symmetric, so there may be a smart attack.
Consider an equivalent problem: What is the minimum possible number of edges in an $n$-vertex graph with no independent set of size 3? Here any pair of vertices corresponds to a member of $A$, any set of 3 vertices corresponds to a member of $B$, and the edges correspond to members of $S$. But consider the complement of this problem: What is the maximum possible number of edges in an $n$-vertex graph with no $K_3$? This is exactly Turán's theorem, so your solution will be:
$$\left|E\left(\overline{T(n,2)}\right)\right| = \left \lfloor \frac{(n-1)^2}{4} \right \rfloor$$