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Let $F$ be an $n$-vertex forest. We say that an edge $e\notin F$ is in theshadow 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 resultsin dimension $d>1$ turn out to be remarkably different from the case in graphs.In particular the corresponding bounds depend on the underlying field ofcoefficients. We find the (tight) analogous theorems for $d=2$. We construct$2$-dimensional "$\mathbb Q$-almost-hypertrees" (defined below) with an emptyshadow. We also show that the shadow of an "$\mathbb F_2$-almost-hypertree"cannot be empty, and its least possible density is $\Theta(\frac{1}{n})$. Inaddition we construct very large hyperforests with a shadow that is empty overevery 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.

Extremal problems on shadows and hypercuts in simplicial complexes