Topological Ramsey spaces of equivalence relations and a dual Ramsey theorem for countable ordinals

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.