Heritability of K\H{o}nig's Property from finite edge sets

TL;DR

This paper investigates whether K"onig's Property in hypergraphs can be inherited from finite subsets of edges, providing a negative answer and exploring related weaker properties.

Contribution

It demonstrates that K"onig's Property is not inheritable from finite edge subsets and examines similar questions for weaker properties.

Findings

K"onig's Property is not inherited from finite subsets

Analysis of weaker properties related to K"onig's Property

Provides counterexamples to inheritance question

Abstract

A hypergraph $H = (V,E)$ is said to have K\H{o}nig's Property if there is a matching $M\subseteq E$ and $S\subseteq V$ such that $|S \cap e| = 1$ for all $e\in M$, and $S$ is a vertex cover of $H$. Aharoni posed the question whether K\H{o}nig's Property is inheritable from finite subsets of $E$. We provide a negative answer and investigate similar questions for weaker properties.