On $\alpha$-largeness and the Paris-Harrington principle in ${\mathrm{R}\mathrm{C}\mathrm{A}}_0$ and ${\mathrm{R}\mathrm{C}\mathrm{A}}_0^{{\displaystyle \ast }}$

We examine, within ${\mathrm{R}\mathrm{C}\mathrm{A}}_0$, the treatment by Ketonen and Solovay on the use of $\alpha$-largeness for giving an upper bound for the Paris–Harrington principle. This proof works fine in ${\mathrm{R}\mathrm{C}\mathrm{A}}_0^{{\displaystyle \ast }}$ for every fixed standard dimension. We also show how to modify the arguments to work within ${\mathrm{R}\mathrm{C}\mathrm{A}}_0^{{\displaystyle \ast }}$ for unrestricted dimensions. To the author’s knowledge, this is the first time that it is confirmed that the treatment can be done within $\mathrm{E}\mathrm{F}\mathrm{A}$ without some transfinite induction added.