Extremal problems on shadows and hypercuts in simplicial complexes

Abstract

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.