Minimal vertex covers in infinite hypergraphs

TL;DR

This paper investigates conditions under which infinite hypergraphs with certain intersection properties have minimal vertex covers, providing several new theorems for different set and cardinal configurations.

Contribution

It introduces new results on minimal vertex covers in infinite hypergraphs with property C(k,ρ), extending understanding in set-theoretic hypergraph theory.

Findings

Proves minimal vertex cover existence for hypergraphs with nowhere stationary sets.

Establishes bounds for cardinals where minimal vertex covers exist.

Provides conditions for hypergraphs with specific intersection properties.

Abstract

In this paper a hypergraph will be identified with the family of its edges. A hypergraph $\mathcal E$ possesses property $C(k,{\rho})$ iff $|\bigcap \mathcal E'|<{\rho}$ for each $\mathcal E'\in {[\mathcal E]}^{k}$. A vertex set $Y\subset \bigcup\mathcal E$ is a "vertex cover" of $\mathcal E$ iff $E\cap Y\ne \emptyset$ for each $E\in \mathcal E$. A vertex cover $Y$ is "minimal" iff no proper subset of $Y$ is vertex cover. If $A$ is a set and $S$ is a set of cardinals, write $$ {[A]}^{S}=\{B\subset A: |B|\in S\}. $$ If ${\lambda}$ and ${\rho}$ are cardinals, $S$ is a set of cardinals, $k\in {\omega}$, then we write $$\mathbf M({{\lambda}},{S},{k},{{\mu}})\to \mathbf{MinVC} $$ iff every hypergraph $\mathcal E\subset {[{\lambda}]}^{S}$ possessing property $C({k},{{\rho}})$ has a minimal vertex cover. If $S=\{\kappa\}$, then we simply write $\mathbf M({{\lambda}},{\kappa},{k},{{\mu}})\to \mathbf{MinVC}$ for $\mathbf M({{\lambda}},\{{\kappa}\},{k},{{\mu}})\to \mathbf{MinVC}$ A set $S$ of cardinals is "nowhere stationary" iff $S\cap {\alpha}$ is not stationary in ${\alpha}$ for any ordinal ${\alpha}$ with $cf({\alpha})>{\omega}$. Countable sets of cardinals, and sets of successor cardinals are nowhere stationary. In this paper we prove: (1) $\mathbf M({{\lambda}},{S},{2},{k})\to \mathbf{MinVC}$ for each nowhere stationary set $S$ of cardinals and ${\omega}\le {\lambda}$, (2) $\mathbf M({{\lambda}},{{\kappa}} ,{2},{{\rho}})\to \mathbf{MinVC}$ provided ${\rho}<\beth_{\omega}\le {\kappa}\le {\lambda}$, (3) $\mathbf M({{\lambda}},{{\omega}},{r},{k})\to \mathbf{MinVC}$ provided ${\omega}\le {\lambda}$ and $k,r\in {\omega}$, (4) $\mathbf M({{\lambda}},{{\omega}_1},{3},{k})\to \mathbf{MinVC}$ provided ${\omega}_1\le {\lambda}$ and $k\in {\omega}$.