Extremal hypercuts and shadows of simplicial complexes

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 ⌊(n−3)24⌋ edges and this is tight. Equivalently, the largest number of edges in an n-vertex cut is ⌊n24⌋. 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 “F₂-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 F₂-almost-hypertree whose shadow has vanishing density. Several intriguing open questions are mentioned as well.