  We study the reverse mathematics of infinitary extensions of the Hales-Jewett
theorem, due to Carlson and Simpson. These theorems have multiple applications
in Ramsey's theory, such as the existence of finite big Ramsey numbers for the
triangle-free graph, or the Dual Ramsey theorem. We show in particular that the
Open Dual Ramsey theorem holds in $\mathsf{ACA}^{+}_0$.
