Related papers: $\mathsf{RT}_2^2$ does not imply $\mathsf{WKL}_0$

We show that the theory $\mathsf{WKL}^*_0+\mathsf{CAC}$ is polynomially simulated by $\mathsf{RCA}_0^*$ with respect to $\forall\Pi^0_3$ formulas. For the proof, we use the method of forcing interpretations and syntactically simulate a…

We prove that any proof of a $\forall \Sigma^0_2$ sentence in the theory $\mathrm{WKL}_0 + \mathrm{RT}^2_2$ can be translated into a proof in $\mathrm{RCA}_0$ at the cost of a polynomial increase in size. In fact, the proof in…

We show that over the weak base theory $\mathrm{RCA}_0^*$, cohesive Ramsey's theorem for pairs $\mathrm{CRT}^2_2$ implies exponential closure of the definable cut $\mathrm{I}^0_1$, which is the intersection of all $\Sigma^0_1$-definable…

In this paper, we show that $\mathrm{RT}^{2}+\mathsf{WKL}_0$ is a $\Pi^{1}_{1}$-conservative extension of $\mathrm{B}\Sigma^0_3$.

We prove that for an arbitrary subtree $T$ of $2^{<\omega}$ with each element extendable to a path, a given countable class $\mathcal{M}$ closed under disjoint union, and any set $A$, if none of the members of $\mathcal{M}$ strongly…

For any pair of ordinals $\alpha<\beta$, $\sf CA_\alpha$ denotes the class of cylindric algebras of dimension $\alpha$, $\sf RCA_{\alpha}$ denote the class of representable $\sf CA_\alpha$s and $\sf Nr_\alpha CA_\beta$ ($\sf Ra CA_\beta)$…

We study the strength of $\RRT^3_2$, Rainbow Ramsey Theorem for colorings of triples, and prove that $\RCA + \RRT^3_2$ implies neither $\WKL$ nor $\RRT^4_2$. To this end, we establish some recursion theoretic properties of cohesive sets and…

Let $\mathsf{TT}^2_k$ denote the combinatorial principle stating that every $k$-coloring of pairs of compatible nodes in the full binary tree has a homogeneous solution, i.e. an isomorphic subtree in which all pairs of compatible nodes have…

No natural principle is currently known to be strictly between the arithmetic comprehension axiom (ACA) and Ramsey's theorem for pairs (RT^2_2) in reverse mathematics. The tree theorem for pairs (TT^2_2) is however a good candidate. The…

This paper investigates the logical strength of completeness theorems for modal propositional logic within second-order arithmetic. We demonstrate that the weak completeness theorem for modal propositional logic is provable in…

We prove that any equational basis that defines RRA over wRRA must contain infinitely many variables. The proof uses a construction of arbitrarily large finite weakly representable but not representable relation algebras whose "small"…

Let $\mathfrak M=(M,\mathcal X)$ be a model of $\mathsf{RCA}_0+\text{$\Sigma^0_2$-bounding}$ in which $\Sigma^0_2(A)$-induction fails for some $A\in\mathcal X$. We show that (i) if $\mathfrak M$ is a model of the combinatorial principle…

We use a second-order analogy $\mathsf{PRA}^2$ of $\mathsf{PRA}$ to investigate the proof-theoretic strength of theorems in countable algebra, analysis, and infinite combinatorics. We compare our results with similar results in the…

We show that RT(2,4) cannot be proved with one typical application of RT(2,2) in an intuitionistic extension of RCA0 to higher types, but that this does not remain true when the law of the excluded middle is added. The argument uses…

This paper uses the framework of reverse mathematics to investigate the strength of two recurrence theorems of topological dynamics. It establishes that one of these theorems, the existence of an almost periodic point, lies strictly between…

It is established that there exists an absolute constant $c>0$ such that for any finite set $A$ of positive real numbers $$|AA+A| \gg |A|^{\frac{3}{2}+c}.$$ On the other hand, we give an explicit construction of a finite set $A \subset…

We make several observations relating the Lie algebra $\mathfrak{g}_2 \subset \mathfrak{so}(7)$, associative $3$-planes, and $\mathfrak{so}(4)$ subalgebras. Some are likely well-known but not easy to find in the literature, while other…

Le Hou\'erou, Patey and Yokoyama defined a parameterized version of $\alpha$-largeness to prove that $\mathsf{WKL}_0 + \mathsf{RT}^2_2$ is a $\forall \Sigma^0_3$-conservative extension of $\mathsf{RCA}_0 + \mathsf{B}\Sigma^0_2$, where…

We show that Brown's lemma is equivalent to Sigma02-induction over RCA0* and that the finite version of Brown's lemma is provable in RCA0 but not in RCA0*.

We prove that $\RCA + \RRT^3_2 \not\vdash \ACA$ where $\RRT^3_2$ is the Rainbow Ramsey Theorem for 2-bounded colorings of triples. This reverse mathematical result is based on a cone avoidance theorem, that every 2-bounded coloring of pairs…