  We define a collection of topological Ramsey spaces consisting of equivalence
relations on $\omega$ with the property that the minimal representatives of the
equivalence classes alternate according to a fixed partition of $\omega$. To
prove the associated pigeonhole principles, we make use of the left-variable
Hales-Jewett theorem and its extension to an infinite alphabet. We also show
how to transfer the corresponding infinite-dimensional Ramsey results to
equivalence relations on countable limit ordinals (up to a necessary
restriction on the set of minimal representatives of the equivalence classes)
in order to obtain a dual Ramsey theorem for such ordinals.
