Related papers: Strong reductions and combinatorial principles
We study the reverse mathematics and computability-the\-o\-re\-tic strength of (stable) Ramsey's Theorem for pairs and the related principles COH and DNR. We show that SRT$^2_2$ implies DNR over RCA$_0$ but COH does not, and answer a…
A complete analysis is given of the computable reductions that hold between $\mathsf{SRT}^2_2$, $\mathsf{SPT}^2_2$, and $\mathsf{SIPT}^2_2$. In particular, while $\mathsf{D}^2_2\le_{\rm sW}\mathsf{SIPT}^2_2\le_{\rm…
We introduce the definability strength of combinatorial principles. In terms of definability strength, a combinatorial principle is strong if solving a corresponding combinatorial problem could help in simplifying the definition of a…
The $\mathsf{SRT}^2_2$ vs.\ $\mathsf{COH}$ problem is a central problem in computable combinatorics and reverse mathematics, asking whether every Turing ideal that satisfies the principle $\mathsf{SRT}^2_2$ also satisfies the principle…
Informally, a mathematical statement is robust if its strength is left unchanged under variations of the statement. In this paper, we investigate the lack of robustness of Ramsey's theorem and its consequence under the frameworks of reverse…
We introduce and study several notions of computability-theoretic reducibility between subsets of $\omega$ that are "robust" in the sense that if only partial information is available about the oracle, then partial information can be…
An open question in reverse mathematics is whether the cohesive principle, $\COH$, is implied by the stable form of Ramsey's theorem for pairs, $\SRT^2_2$, in $\omega$-models of $\RCA$. One typical way of establishing this implication would…
A relatively new topic in computability theory is the study of notions of computation that are robust against mistakes on some kind of small set. However, despite the recent popularity of this topic relatively foundational questions about…
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…
The enterprise of comparing mathematical theorems according to their logical strength is an active area in mathematical logic. In this setting, called reverse mathematics, one investigates which theorems provably imply which others in a…
The tree theorem for pairs ($\mathsf{TT}^2_2$), first introduced by Chubb, Hirst, and McNicholl, asserts that given a finite coloring of pairs of comparable nodes in the full binary tree $2^{<\omega}$, there is a set of nodes isomorphic to…
We study the positions in the Weihrauch lattice of parallel products of various combinatorial principles related to Ramsey's theorem. Among other results, we obtain an answer to a question of Brattka, by showing that Ramsey's theorem for…
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…
This paper continues to study the connection between reverse mathematics and Weihrauch reducibility. In particular, we study the problems formed from Maltsev's theorem on the order types of countable ordered groups. Solomon showed that the…
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…
We study a restriction of Ramsey's theorem for 2-coloring of triples, in which homogeneous sets for color~1 are of bounded size ($\mathsf{BRT}^3_2$). We prove that the computational content of this statement is very close to Ramsey's…
The stable Ramsey's theorem for pairs has been the subject of numerous investigations in mathematical logic. We introduce a weaker form of it by restricting from the class of all stable colorings to subclasses of it that are non-null in a…
The paper is devoted to a reverse-mathematical study of some well-known consequences of Ramsey's theorem for pairs, focused on the chain-antichain principle $\mathsf{CAC}$, the ascending-descending sequence principle $\mathsf{ADS}$, and the…
We identify computability-theoretic properties enabling us to separate various statements about partial orders in reverse mathematics. We obtain simpler proofs of existing separations, and deduce new compound ones. This work is part of a…
In this thesis, we investigate the computational content and the logical strength of Ramsey's theorem and its consequences. For this, we use the frameworks of reverse mathematics and of computable reducibility. We proceed to a systematic…