Random reals and the relation \(\omega_ 1\to (\omega_ 1,(\alpha:n))^ 2\) (Q1893262)

The paper is devoted to the proof of the following result: Assume that \(\text{MA}_{\aleph_1}\) holds in the universe, and let \(B\) denote an atomless measure algebra via a countably additive measure \(\mu: B\to [0,1 ]\). Then any extension obtained by forcing with \(B\) over the universe satisfies the following: Given \(F: [\omega_1 ]^2\to \{ 0,1 \}\), \(\alpha< \omega_1\), and \(n< \omega\), either \(\omega_1\) can be partitioned into \(P_m\), \(m\in \omega\), so that for each \(m\), \(F\) is identically 0 on \([P_m ]^2\), or else there are \(C\in [\omega_1 ]^\alpha\) and \(D\in [\{ \delta\in \omega_1 : \forall \gamma\in C\;\gamma< \delta\} ]^n\) such that \(F\) is identically \(1\) on \(C\times D\). (We recall that given an ordinal \(\beta\) and a well-ordered set \(X\), \([X]^\beta\) denotes the collection of all order-type \(\beta\) subsets of \(X\)).