The authors study the collection \(C\) of all infinite cardinals \(\kappa\) with the property that for every \(F : \bigcup_{n \in \omega} {}^n\kappa \to 2\), there are \(H_i \subseteq \kappa\) for \(i \in \omega\) such that (a) \(|H_i|= 2\) for each \(i \in \omega\), and (b) \(F\) is constant on \(\prod^n_{i = 0} H_i\) for every \(n \in \omega\). It is immediate that if \(\kappa\) lies in \(C\), then so does every larger cardinal. Hence the main problems are (1) to determine whether \(C\) has any element, and (2) to describe the least element of \(C\) in case \(C\) is nonempty. The authors also consider larger collections obtained by replacing (a) by \((\text{a}')\): \(|H_i|= 2\) for infinitely many \(i\), and (b) by \((\text{b}')\): \(F\) is constant on \(\prod^n_{i = 0} H_i\) for infinitely many \(n\).