Constructions and Bounds for Splitting and Separating Systems

Abstract

Suppose $m$ and $t$ are integers such that $0<t\le m$. An $(m,t)$-splitting system is a pair $(X,\mathcal{B})$ that satisfies for every $Y\subseteq X$ with $|Y|=t$, there is a subset $B$ of $X$ in $\mathcal{B}$, such that $|B\cap Y|=\lfloor \frac{t}{2}\rfloor$ or $|(X\setminus B)\cap Y|=\lceil \frac{t}{2}\rceil$. Suppose $m$, $t_1$, and $t_2$ are integers such that $t_1+t_2\le m$. An $(m,t_1,t_2)$-separating system is a pair $(X,\mathcal{B})$ which satisfies for every $P\subseteq X$, $Q\subseteq X$ with $|P|=t_1$, $|Q|=t_2$, and $P\cap Q=\mathrm{\varnothing }$, there exists a block $B\in \mathcal{B}$ for which either $P\subseteq B$, $Q\cap B=\mathrm{\varnothing }$ or $Q\subseteq B$, $P\cap B=\mathrm{\varnothing }$. We will give some results on splitting systems and separating systems for $t=5$ and $t=6$.