Effect of a random real on \(\kappa\to \left(\kappa,{\alpha \choose \omega_ 1}\right)^ 2\) (Q1346388)

In this paper the author deals with the partition relation \(\kappa\to \left( \kappa, \left( \begin{smallmatrix} \alpha\\ \omega_ 1\end{smallmatrix} \right) \right)^ 2\). It is well known that this partition relation follows from \(\text{MA}_ \kappa\) for \(\alpha< \omega_ 1\) and \(\text{cf} (\kappa) >\omega\). Further on it is known that adding a Cohen or random real causes \(\text{MA}_ \kappa\) to fail. Here the author shows that this partition relation does not hold in \({\mathcal M} [r ]\) for \(r\) a single random real, \(\alpha< \omega_ 1\), \(\text{cf} (\kappa)> \omega\) and \({\mathcal M} \models \text{MA}_ \kappa\).