Bisecting and D-secting families for set systems

Abstract: Let $n$ be any positive integer and $\mathcal{F}$ be a family of subsets of $[n]$. A family $\mathcal{F}'$ is said to be $D$-\emph{secting} for $\mathcal{F}$ if for every $A \in \mathcal{F}$, there exists a subset $A' \in \mathcal{F}'$ such that $|A \cap A'| - |A \cap ([n] \setminus A')|=i$, where $i \in D$, $D \subseteq {-n,-n+1,\ldots,0,\ldots,n}$. A $D$-\emph{secting} family $\mathcal{F}'$ of $\mathcal{F}$, where $D={-1,0,1}$, is a \emph{bisecting} family ensuring the existence of a subset $A' \in \mathcal{F}'$ such that $|A \cap A'| \in {\lceil \frac{|A|}{2}\rceil,\lfloor \frac{|A|}{2}\rfloor}$, for each $A \in \mathcal{F}$. In this paper, we study $D$-secting families for $\mathcal{F}$ with restrictions on $D$, and the cardinalities of $\mathcal{F}$ and the subsets of $\mathcal{F}$.