Negative partition relations for uncountable cardinals of cofinality \(\omega\) (Q2755004)

If \(\lambda\) is an uncountable cardinal of cofinality \(\omega\), Baumgartner showed that \(I^+_{\omega,\lambda}\to (I^+_{\omega,\lambda})^2\) does not hold in case \(\mu^{\aleph_0}<\lambda\) for every cardinal \(\mu<1\), see \textit{Alives} [Partitions of finite substructures, PhD Thesis, Pennsylvania State Univ. (1985)]. The argument has been modified in this paper to prove that \(I^+_{\omega,\omega_\omega}\not\to(I^+_{\omega,\omega_\omega})^2\) in case \(\partial <\omega_\omega\).NEWLINENEWLINENEWLINE(If \(\nu,\mu\) are cardinals, \(P_\nu(\mu)=\{a\subseteq \mu:|a|<\nu\}\) and \(\widehat a=\{b\in P_\nu(\mu):a\subseteq b\}\) for \(a \in P_\nu(\mu)\). \(I_{\nu,\mu}\) consists of all \(A\subseteq P_\nu(\mu)\) such that \(A\cap \widehat a=\phi\) for some \(a \in P_\nu (\mu)\). Also \(I_{\nu,\mu}^+=P(P_\nu(\mu))-I_{\nu,\mu}\).NEWLINENEWLINENEWLINEFurther \(I_{\omega,\lambda}^+\to (I^+_{\omega,\lambda})^2\) means that given \(F:P_\omega(\lambda)\times P_\omega(\lambda)\to 2\) and \(A\in I^+_{\omega,\lambda}\), there is \(B{\i}I^+_{\omega,\lambda}\cap P(A)\) such that \(F\) is constant on \(\{(a,b)\in A\times B:a\subset b\}\). The dominating number \(\partial\) is the least cardinality of any \(G\subseteq \omega_\omega\) with the property that for every \(f\in\omega_\omega\), there is \(g\in G\) such that \(f(n)\leq g(n)\) for all \(n\in \omega\).