A note on a set-mapping problem of Hajnal and Máté (Q1329219)

The authors consider set mappings \(F\) with domain the set \([\omega_ 2]^ 2\) of all pairs from the ordinal \(\omega_ 2\) and range a subset of \([\omega_ 2]^{<\lambda}\) for some cardinal \(\lambda\) (where \([\omega_ 2]^{<\lambda}=\{ S\subseteq\omega_ 2\); \(| S|< \lambda\}\). A set \(Y\subseteq\omega_ 2\) is said to be free for \(F\) if \(\gamma\notin F(\{\alpha,\beta\})\) for all distinct triples \(\alpha\), \(\beta\), \(\gamma\) from \(Y\). We shall say the set mapping \(F\) is restricted if \(F(\{\alpha,\beta\})\subseteq\{\gamma\); \(\alpha<\gamma<\beta\}\) whenever \(\alpha<\beta< \omega_ 2\). A result of \textit{A. Hajnal} and \textit{A. Máté} [Logic Colloq. '73, Proc. Bristol 1973, 347-379 (1975; Zbl 0324.04004)] shows that every restricted set mapping \(f: [\omega_ 2]^ 2\to [\omega_ 2]^{<\aleph_ 1}\) has an infinite free set. They asked whether every restricted set mapping \(f: [\omega_ 2]^ 2\to [\omega_ 2]^{<\aleph_ 0}\) has an uncountable free set. The paper under review uses models of \textit{U. Abraham} and the second author [J. Symb. Logic 51,180-189 (1986; Zbl 0633.03042)] to show that it is consistent that there exists a restricted set mapping \(f: [\omega_ 2]^ 2\to [\omega_ 2]^{<\aleph_ 1}\) with no uncountable free set, and they deduce from this that it is consistent that the answer to Hajnal and Máté's question is `no', by constructing a restricted set mapping \(f: [\omega_ 2]^ 2\to [\omega_ 2]^{<2}\) with no uncountable free set.