[For the entire collection see Zbl 0559.00001.]
For present purposes, an almost disjoint (AD) family on a set X is a set \({\mathcal A}\subseteq [X]^{\omega}\) such that \(| A\cap A'| <\omega\) whenever A, A’ are distinct elements of \({\mathcal A}\); a MAD family is a maximal AD family. For \({\mathcal M}\subseteq {\mathcal P}(X)\), an AD refinement (abbreviated ADR) for \({\mathcal M}\) is an AD family \({\mathcal A}\subseteq [X]^{\omega}\) such that for each \(M\in {\mathcal M}\) there is \(A\in {\mathcal A}\) such that \(A\subseteq M\). For \({\mathcal A}\) an AD family on (the cardinal number) \(\tau\), the authors write \[ {\mathcal J}^+({\mathcal A})=\{Y\in [\tau]^{\omega}: | \{A\in {\mathcal A}: | A\cap Y| =\omega \}| \geq \omega \}. \] For \({\mathcal A},{\mathcal B}\subseteq {\mathcal P}(X)\), the notation \({\mathcal B}\prec {\mathcal A}\) means that for all \(B\in {\mathcal B}\) there is \(A\in {\mathcal A}\) such that \(B\subseteq A.\)
The statement GH(\(\tau)\) is the statement that for every MAD family \({\mathcal A}\) on \(\tau\) there is a MAD family \({\mathcal B}\) on \(\tau\) such that (1) \({\mathcal B}\prec {\mathcal A}\) and (2) if \(Y\in {\mathcal J}^+({\mathcal B})\) then there is \(B\in {\mathcal B}\) such that \(| B\setminus Y| <\omega\). Statement GH(\(\omega)\) is equivalent to Hechler’s conjecture, that is, the statement that each nowhere dense subspace Z of \(\beta(\omega)\setminus \omega\) is a \({\mathfrak c}\)-set (in the sense that there is an AD family \(\{A_{\xi}:\) \(\xi <{\mathfrak c}\}\) on \(\omega\) with \(Z\subseteq cl_{\beta (\omega)}A_{\xi}\) for each \(\xi <{\mathfrak c}).\)
The authors prove several theorems of combinatorial type. Among their results (in ZFC) are: (1) GH(\(\omega)\) iff GH(\({\mathfrak c})\); (2) for each \(\tau\geq \omega\), GH(\(\tau)\) iff \(GH(\tau^+)\); (3) if \(X\subseteq \beta (\omega)\setminus \omega\) with \(| X| <{\mathfrak c}\), then \(\cup X\) has an ADR; (4) if \({\mathcal A}\) is a MAD family on \(\omega\) with \(| {\mathcal A}| =\aleph_ 1\), then \({\mathcal J}^+({\mathcal A})\) has an ADR. Statement GH(\(\omega)\) remains unsettled in ZFC, but the authors find some special families M for which an ADR exists. Examples: (a) \({\mathcal M}=\{M\in [\omega]^{\omega}:\) lim sup \(| M\cap [0,n)| /n>0\}\); (b) \({\mathcal M}=\{M\subseteq {\mathbb{R}}:\) M has infinitely many accumulation points}.