Extremal problems on shadows and hypercuts in simplicial complexes

Understand

Let $F$ be an $n$-vertex forest.

• We say that an edge $e\notin F$ is in the shadow of $F$ if $F\cup\{e\}$ contains a cycle.
• It is easy to see that if $F$ is "almost a tree", that is, it has $n-2$ edges, then at least $\lfloor\frac{n^2}{4}\rfloor$ edges are in its shadow and this is tight.
• Equivalently, the largest number of edges an $n$-vertex cut can have is $\lfloor\frac{n^2}{4}\rfloor$.