Note as to inclusion-minimal non-Bondy systems

TL;DR

This paper characterizes the existence of inclusion-minimal non-Bondy systems of a given size on a finite set, establishing precise bounds for their sizes based on the set's cardinality.

Contribution

It provides a complete characterization of when inclusion-minimal non-Bondy systems exist for a given size and set, filling a gap in the understanding of these combinatorial structures.

Findings

Inclusion-minimal non-Bondy systems exist if and only if their size t satisfies s+1 ≤ t ≤ 2s.

The bounds for the size of such systems are exactly determined by the size of the underlying set.

The result applies to any finite set with at least 6 elements.

Abstract

Let $S$ be a finite set, $s=|S|\ge6$. Given a non-negative integer $t$, there exists an inclusion-minimal non-Bondy system $\mathscr{A}$ of size $t$ on $S$ if and only if $s+1\le t\le2s$.