Minimal vertex covers in infinite hypergraphs

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 $|\bigcap {\mathcal{E}}^{\prime }|<\rho$ for each ${\mathcal{E}}^{\prime }\in {[\mathcal{E}]}^k$. A vertex set $Y\subset \bigcup \mathcal{E}$ is a "vertex cover" of $\mathcal{E}$ iff $E\cap Y\ne \varnothing$ 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 $k\in \omega$.

13 pages