A Hall-type theorem for triplet set systems based on medians in trees

TL;DR

This paper extends Hall's theorem by characterizing collections of triplet subsets of a set through median conditions in trees, providing new insights into set systems and their representations.

Contribution

It introduces a novel characterization of triplet set systems using median conditions in trees, extending classical Hall-type theorems to more complex set collections.

Findings

Characterization of triplet systems via median conditions in trees

Extension of the result to arbitrary subset sizes

New combinatorial criteria for set system representations

Abstract

Given a collection $\C$ of subsets of a finite set $X$, let $\bigcup \C = \cup_{S \in \C}S$. Philip Hall's celebrated theorem \cite{hall} concerning `systems of distinct representatives' tells us that for any collection $\C$ of subsets of $X$ there exists an injective (i.e. one-to-one) function $f: \C \to X$ with $f(S) \in S$ for all $S \in \C$ if and and only if $\C$ satisfies the property that for all non-empty subsets $\C'$ of $\C$ we have $|\bigcup \C'| \geq |\C'|$. Here we show that if the condition $|\bigcup \C'| \geq |\C'|$ is replaced by the stronger condition $|\bigcup \C'| \geq |\C'|+2$, then we obtain a characterization of this condition for a collection of 3-element subsets of $X$ in terms of the existence of an injective function from $\C$ to the vertices of a tree whose vertex set includes $X$ and that satisfies a certain median condition. We then describe an extension of this result to collections of arbitrary-cardinality subsets of $X$.