We study the first-order consequences of Ramsey's Theorem for $k$-colourings of $n$-tuples, for fixed $n, k \ge 2$, over the relatively weak second-order arithmetic theory $\mathrm{ RCA}^*_0$. Using the Chong-Mourad coding lemma, we show that in a model of $\mathrm{ RCA}^*_0 + \neg \mathrm{ I}\Sigma^0_1$, $\mathrm{ RT}^n_k$ is equivalent to its own relativization to any proper $\Sigma^0_1$-definable cut, so its truth value remains unchanged in all extensions of the model with the same first-order universe. We give an axiomatization of the first-order consequences of $\mathrm{ RCA}^*_0 + \mathrm{ RT}^n_k$ for $n \ge 3$. We show that they form a non-finitely axiomatizable subtheory of PA whose $\Pi_3$ fragment is $\mathrm{ B}\Sigma_1 + \exp$ and whose $\Pi_{ \ell+3}$ fragment for $\ell \ge 1$ lies between $\mathrm{ I}\Sigma_\ell \Rightarrow \mathrm{ B}\Sigma_{ \ell+1}$ and $\mathrm{ B}\Sigma_{ \ell+1}$. We also consider the first-order consequences of $\mathrm{ RCA}^*_0 + \mathrm{ RT}^2_k$. We show that they form a subtheory of $\mathrm{ I}\Sigma_2$ whose $\Pi_3$ fragment is $\mathrm{ B}\Sigma_1 + \exp$ and whose $\Pi_4$ fragment is strictly weaker than $\mathrm{ B}\Sigma_2$ but not contained in $\mathrm{ I}\Sigma_1$. Additionally, we consider a principle $\Delta^0_2$-$\mathrm{ RT}^2_2$, defined like $\mathrm{ RT}^2_2$ but with both the $2$-colourings and the solutions allowed to be $\Delta^0_2$-sets. We show that the behaviour of $\Delta^0_2$-$\mathrm{ RT}^2_2$ over $\mathrm{ RCA}_0 + \mathrm{ B}\Sigma^0_2$ is similar to that of $\mathrm{ RT}^2_2$ over $\mathrm{ RCA}^*_0$, and that $\mathrm{ RCA}_0 + \mathrm{ B}\Sigma^0_2 + \Delta^0_2$-$\mathrm{ RT}^2_2$ is $\Pi_4$- but not $\Pi_5$-conservative over $\mathrm{ B}\Sigma_2$. However, the statement we use to witness lack of $\Pi_5$-conservativity is not provable in $\mathrm{ RCA}_0 +\mathrm{ RT}^2_2$.
Categories: math.LO