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Note as to size-minimal hypercompletly separating systems

B. Batikova, T. J. Kepka, P. C. Nemec

Abstract

If $S$ is a non-empty finite set, $|S|=s$, then a system $\mathscr{A}$ of subsets of $S$ is a size-minimal hypercompletely separable system (i.e., for every $a\in S$ there are $A,B\in\mathscr{A}$ such that $A\cap B=\{a\}$) if and only if $|\mathscr{A}|=\left\lceil\frac{1+\sqrt{8s+1}}2\right\rceil$.

Note as to size-minimal hypercompletly separating systems

Abstract

If is a non-empty finite set, , then a system of subsets of is a size-minimal hypercompletely separable system (i.e., for every there are such that ) if and only if .
Paper Structure (4 sections, 7 theorems)

This paper contains 4 sections, 7 theorems.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Preparatory results
  4. Main results

Key Result

Lemma 3.1

Let $k$ be a non-negative integer. Then: (i) $\tau(\alpha(k))=k+1$ . (ii) If $t$ is an integer such that $\alpha(k)+1\le t\le\alpha(k+1)$ then $\tau(t)=k+2$. (iii) $\tau(k)=\frac{1+\sqrt{8k+1}}{2}$ if and only if $k=\alpha(l)$ for some $l\ge0$. (iv) $k\ge\tau(k)$ for $k\ge3$, and $k>\tau(k)$ for $k\

Theorems & Definitions (16)

  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.4
  • proof
  • Proposition 3.5
  • proof
  • Lemma 3.7
  • proof
  • ...and 6 more