The cohesive and stable Ramsey theorems and proof size over a weak base theory

We show that over the weak base theory ${\mathrm{R}\mathrm{C}\mathrm{A}}_0^{\ast }$, cohesive Ramsey’s theorem for pairs ${\mathrm{C}\mathrm{R}\mathrm{T}}_2^2$ implies exponential closure of the definable cut ${\mathrm{I}}_1^0$, which is the intersection of all ${\mathrm{\Sigma }}_1^0$-definable cuts. Consequences include non-elementary proof speedup of ${\mathrm{R}\mathrm{C}\mathrm{A}}_0^{\ast }+{\mathrm{C}\mathrm{R}\mathrm{T}}_2^2$ over ${\mathrm{R}\mathrm{C}\mathrm{A}}_0^{\ast }$ for ${\mathrm{\Pi }}_1$ sentences and the unprovability of ${\mathrm{C}\mathrm{R}\mathrm{T}}_2^2$ in ${\mathrm{R}\mathrm{C}\mathrm{A}}_0^{\ast }+\mathrm{C}\mathrm{A}\mathrm{C}$. On the other hand, we show that ${\mathrm{R}\mathrm{C}\mathrm{A}}_0^{\ast }+{\mathrm{S}\mathrm{R}\mathrm{T}}_2^2$, where ${\mathrm{S}\mathrm{R}\mathrm{T}}_2^2$ is stable Ramsey’s theorem for pairs, is polynomially simulated by ${\mathrm{R}\mathrm{C}\mathrm{A}}_0^{\ast }$ with respect to proofs of $\mathrm{\forall }{\mathrm{\Pi }}_3^0$ sentences. Nevertheless, ${\mathrm{S}\mathrm{R}\mathrm{T}}_2^2$ also implies a nontrivial property of ${\mathrm{I}}_1^0$, specifically closure under functions of quasipolynomial growth rate.