The reverse mathematics of the Ordered Variable Word theorem

In this article, we study the reverse mathematics of variable word theorems used in the proof of the Dual Ramsey theorem. We prove that the Ordered Variable Word theorem does not imply Ramsey’s theorem for pairs and that every computable instance admits a solution of low${}_2$ degree. This is used to prove the Carlson-Simpson Lemma and the Open Dual Ramsey theorem over~${\mathsf{A}\mathsf{C}\mathsf{A}}_0$, thereby answering some 40-years old open questions. These results have consequences in the reverse mathematics of structural Ramsey theory.