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 $\mathrm{RCA}_0 +\mathrm{RT}^2_2$.

How strong is Ramsey's theorem if infinity can be weak?

CAC for trees is the statement asserting that any infinite subtree of $\mathbb{N}^{<\mathbb{N}}$ has an infinite path or an infinite antichain. In this paper, we study the computational strength of this theorem from a reverse mathematical viewpoint. We prove that TAC for trees is robust, that is, there exist several characterizations, some of which already appear in the literature, namely, the tree antichain theorem (TCAC) introduced by Conidis, and the statement SHER introduced by Dorais et al. We show that CAC for trees is computationally very weak, in that it admits probabilistic solutions.

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The reverse mathematics of cac for trees

We study the pigeonhole principle for $\Sigma_2$-definable injections with domain twice as large as the codomain, and the weak Konig lemma for $\Delta^0_2$-definable trees in which every level has at least half of the possible nodes. We show that the latter implies the existence of $2$-random reals, and is conservative over the former. We also show that the former is strictly weaker than the usual pigeonhole principle for $\Sigma_2$-definable injections.

Where Pigeonhole Principles meet K\"onig Lemmas

Prior work of Gavryushkin, Khoussainov, Jain and Stephan investigated what algebraic structures can be realised in worlds given by a positive (= recursively enumerable) equivalence relation which partitions the natural numbers into infinitely many equivalence classes. The present work investigates the infinite one-one numbered recursively enumerable (r.e.) families realised by such relations and asks how the choice of the equivalence relation impacts the learnability properties of these classes when studying learnability in the limit from positive examples, also known as learning from text. For all choices of such positive equivalence relations, for each of the following entries, there are one-one numbered r.e. families which satisfy it: (a) they are behaviourally correctly learnable but not vacillatorily learnable; (b) they are explanatorily learnable but not confidently learnable; (c) they are not behaviourally correctly learnable. Furthermore, there is a positive equivalence relation which enforces that (d) every vacillatorily learnable one-one numbered family of languages closed under this equivalence relation is already explanatorily learnable and cannot be confidently learnable.

Learnability and Positive Equivalence Relations

We study the pigeonhole principle for Σ2-definable injections with domain twice as large as the codomain, and the weak Konig lemma for Δ02-definable trees in which every level has at least half of the possible nodes. We show that the latter implies the existence of 2-random reals, and is conservative over the former. We also show that the former is strictly weaker than the usual pigeonhole principle for Σ2-definable injections.

https://biblio.ugent.be/publication/8669009/file/8669010.pdf

David Belanger

Papers

Where Pigeonhole Principles meet K\"onig Lemmas

Learnability and Positive Equivalence Relations

Learnability and Positive Equivalence Relations

Where pigeonhole principles meet König lemmas