Bisecting and D-secting families for set systems

Let $n$ be any positive integer and $\mathcal{F}$ be a family of subsets of $[n]$. A family ${\mathcal{F}}^{\mathrm{\prime }}$ is said to be $D$-\emph{secting} for $\mathcal{F}$ if for every $A\in \mathcal{F}$, there exists a subset $A^{\mathrm{\prime }}\in {\mathcal{F}}^{\mathrm{\prime }}$ such that $\mathrm{\mid }A\cap A^{\mathrm{\prime }}\mathrm{\mid }-\mathrm{\mid }A\cap ([n]\setminus A^{\mathrm{\prime }})\mathrm{\mid }=i$, where $i\in D$, $D \subseteq {-n,-n+1,\ldots,0,\ldots,n}$. A $D$-\emph{secting} family ${\mathcal{F}}^{\mathrm{\prime }}$ of $\mathcal{F}$, where $D=\{-1,0,1\}$, is a \emph{bisecting} family ensuring the existence of a subset $A^{\mathrm{\prime }}\in {\mathcal{F}}^{\mathrm{\prime }}$ such that $|A \cap A’| \in {\lceil \frac{|A|}{2}\rceil,\lfloor \frac{|A|}{2}\rfloor}$,foreach$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 $\mathcal{F}$.