# Extremal hypercuts and shadows of simplicial complexes

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## Abstract

Let *F* be an *n*-vertex forest. An edge *e* ∉ *F* is said to be in *F*’s shadow if *F* ∪ {*e*} contains a cycle. It is easy to see that if *F* is an “almost tree”, i.e., a forest that contains two components, then its shadow contains at least \(\left\lfloor {\frac{{{{(n - 3)}^2}}}{4}} \right\rfloor \) edges and this is tight. Equivalently, the largest number of edges in an *n*-vertex cut is \(\left\lfloor {\frac{{{n^2}}}{4}} \right\rfloor \). These notions have natural analogs in higher *d*-dimensional simplicial complexes which played a key role in several recent studies of random complexes. The higher-dimensional situation differs remarkably from the one-dimensional graph-theoretic case. In particular, the corresponding bounds depend on the underlying field of coefficients. In dimension *d* = 2 we derive the (tight) analogous theorems. We construct 2-dimensional “ℚ-almost-hypertrees” (defined below) with an empty shadow. We prove that an “\(\mathbb{F}_2\)
-almost-hypertree” cannot have an empty shadow, and we determine its least possible size. We also construct large hyperforests whose shadow is empty over every field. For *d* ≥ 4 even, we construct a *d*-dimensional \(\mathbb{F}_2\)
-almost-hypertree whose shadow has vanishing density.

Several intriguing open questions are mentioned as well.

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