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

arXiv comment15 pages, text overlap with https://www.ams.org/journals/tran/1987-300-01/S0002-9947-1987-0871675-6/S0002-9947-1987-0871675-6.pdf in Theorem 7