SEPARATION NUMBERS WITH RESPECT TO SQUARE NUMBERS

Abstract

A partially ordered set $P=(X, \prec)$ is 'top $r$ separable' if its ground set can be partitioned as $X=U \cup(X \backslash U)$ so that $v \prec u$ for all $u \in U$ and all $v \in X \backslash U$ with $|U|=r$. For a given $n$-set, the minimum number of $k$-subsets whose transitive closure of chains on each subsets is top $r$-separable, denoted by $S(n, k, r)$ is called the separation number of $n, k$ and $r$. In this paper, we first give some basic properties of $S(n, k, r)$, and obtain that $S(5,3,2)=3$ and $S(4,2,2)=4$. Next, we prove that $S\left(m^2, m, 1\right)=1$ and $S\left(m^2, m, r\right)$ $=m+2$ if $2 \leq r \leq\left\lfloor\frac{-1+\sqrt{8 m+9}}{2}\right\rfloor$. In addition, this result can be seen as $S\left(\left(T_r-1\right)^2, T_r-1, r\right)=T_r+1$, in terms of triangular number $T_r$ of order a natural number $r$.

Received: November 6, 2021  
Accepted: January 3, 2022