Martin's axiom and separated mad families
Two families of subsets of \(\omega\) are said to be separated if there is a set which almost contains every element of one of the families, while it is almost disjoint from every element of the other family. An almost disjoint family is said to be \(\omega_1\)-separated if any disjoint pair of \(\leq\omega_1\)-sized subsets are separated. A maximal almost disjoint family \(\mathcal{A}\subseteq [\omega]^\omega\) is \(\omega_1\)-special if for every subfamily \(\mathcal{A^\prime}\in [\mathcal{A}]^{\leq \omega_1}\) there is a function \(c:[\mathcal{A}^\prime]^{<\omega}\to\omega\) and a linear ordering \(<\) of \(\mathcal{A}^\prime\) such that for each \(n\in\omega\) and \(n\)-tuples \(B_1<\cdots<B_n\), \(C_1,\cdots<C_n\) from \(\mathcal{A}^\prime\) if \(c({B_1,\cdots,B_n})= c({C_1,\cdots,C_n})=k\) then for all \(i\neq j\) the intersection \(B_i\cap C_j\) is contained in \(k\). Two families of subsets of \(\omega\) are said to be orthogonal if each element of one of the families is almost disjoint from each element of the other family. In [Izv. Akad. Nauk SSSR, Ser. Mat. 11, 403--410 (1947; Zbl 0029.34703)], \textit{N. N. Luzin} constructed an orthogonal pair of \(\aleph_1\)-sized almost disjoint subsets of \(\omega\) which can not be separated. In [\textit{A. Dow}, Can. Math. Bull. 54, No. 2, 270--276 (2011; Zbl 1244.54064)], it is shown that PFA implies that every maximal almost disjoint family contains a Luzin sequence, and so an \(\omega_1\)-sized subfamily, which has the property that each of its uncountable pairs cannot be separated. In the present paper the authors obtain a model of MA in which \(\mathfrak{c}=\omega_2\) and there is an \(\omega_1\)-separated maximal almost disjoint family. The proof uses a result of Silver according to which \(\text{MA}_{\omega_1}\) implies that every \(\omega_1\)-special family is \(\omega_1\)-separated. In order to obtain their main result, using a finite support iteration of ccc posets of length \(\omega_2\), the authors force a maximal almost disjoint family \(\mathcal{A}\) which is \(\omega_1\)-special and \(\omega_2\)-indestructible, i.e., remains maximal after forcing with any poset of size less than \(\omega_2\). Over the resulting model they obtain MA in a standard way via a finite support iteration of ccc posets of size \(\leq \omega_1\). Thus, in the final model, by Silver's result, \(\mathcal{A}\) is \(\omega_1\)-separated.
scientific article; zbMATH DE number 6113990