Independence of Ramsey theorem variants using $\varepsilon _0$
Harvey M. Friedman, Florian Pelupessy · Proceedings of the American Mathematical Society · 2015
We show that Friedman’s finite adjacent Ramsey theorem is unprovable in Peano Arithmetic, and give a new proof of the unprovability of the Paris–Harrington theorem in $\mathrm {PA}$. We also determine the status of these theorems for each dimension. It is to be noted that the finite adjacent Ramsey theorem for dimension $d$ is equivalent to the Paris–Harrington theorem for dimension $d+1$.