  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
$\lfloor\frac{n^2}{4}\rfloor$. These notions have natural analogs in higher
$d$-dimensional simplicial complexes, graphs being the case $d=1$. The results
in dimension $d>1$ turn out to be remarkably different from the case in graphs.
In particular the corresponding bounds depend on the underlying field of
coefficients. We find the (tight) analogous theorems for $d=2$. We construct
$2$-dimensional "$\mathbb Q$-almost-hypertrees" (defined below) with an empty
shadow. We also show that the shadow of an "$\mathbb F_2$-almost-hypertree"
cannot be empty, and its least possible density is $\Theta(\frac{1}{n})$. In
addition we construct very large hyperforests with a shadow that is empty over
every field.
  For $d\ge 4$ even, we construct $d$-dimensional $\mathbb{F}
_2$-almost-hypertree whose shadow has density $o_n(1)$.
  Finally, we mention several intriguing open questions.
