Reverse mathematics and Ramsey's property for trees

Corduan, Jared; Groszek, Marcia J.; Mileti, Joseph R.

doi:10.2178/jsl/1278682209

Cited by 19 publications

(22 citation statements)

References 4 publications

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“…This Erdős-Rado theorem shares an essential feature with another strengthening of Ramsey's theorem for pairs already studied in reverse mathematics: the tree theorem for pairs [2,3,5,15]. Both TT 2 2 and (ℵ 0 , η) 2 lie between the arithmetic comprehension axiom and RT 2 2 , but more than that, they share a disjoint extension commitment.…”

Section: Discussion and Questionsmentioning

confidence: 63%

“…One may think of BΣ 0 2 as asserting that the finite union of finite sets is finite (see for instance [7]). We show that (η) 1 <∞ , the corresponding pigeonhole principle for rationals, is strictly stronger than BΣ 0 2 , and hence has the same reverse mathematics status as the tree theorem for singletons (TT 1 ) [3].…”

Section: Introductionmentioning

confidence: 77%

“…However, no natural statement besides Ramsey's theorem for pairs (RT 2 ) is known to be strictly between ACA 0 and RT 2 2 over RCA 0 . The only known candidate is the tree theorem for pairs (TT 2 2 ) studied in [2,3,5,15]. We show that (ℵ 0 , η) 2 also lies between ACA 0 and RT 2 2 , and so represents another candidate, arguably more natural than TT 2 2 .…”

Section: Introductionmentioning

confidence: 91%

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Coloring the rationals in reverse mathematics

Frittaion

Patey

2017

COM

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Ramsey's theorem for pairs asserts that every 2-coloring of the pairs of integers has an infinite monochromatic subset. In this paper, we study a strengthening of Ramsey's theorem for pairs due to Erdős and Rado, which states that every 2-coloring of the pairs of rationals has either an infinite 0-homogeneous set or a 1-homogeneous set of order type η, where η is the order type of the rationals. This theorem is a natural candidate to lie strictly between the arithmetic comprehension axiom and Ramsey's theorem for pairs. This Erdős-Rado theorem, like the tree theorem for pairs, belongs to a family of Ramsey-type statements whose logical strength remains a challenge.

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Section: Discussion and Questionsmentioning

confidence: 63%

Section: Introductionmentioning

confidence: 77%

Section: Introductionmentioning

confidence: 91%

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Coloring the rationals in reverse mathematics

Frittaion

Patey

2017

COM

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“…From the known proof that the rationals do not have the 2, 2-Ramsey property, we can derive the following fact [3]. Proposition 3.1.…”

Section: Countable Chain-ramsey Partial Orderingsmentioning

confidence: 97%

“…Fouché [5] investigates the finite combinatorics of this proposition. Corduan, Groszek, and Mileti [3] show that if the partial ordering P is a countably infinite rooted tree, then P is chain-Ramsey if and only if P is biembeddable with either the natural numbers or the infinite binary tree, both with the usual orderings.…”

Section: Introductionmentioning

confidence: 99%

Ramsey Properties of Countably Infinite Partial Orderings

Groszek

2013

Electron. J. Combin.

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A partial ordering $\mathbb P$ is chain-Ramsey if, for every natural number $n$ and every coloring of the $n$-element chains from $\mathbb P$ in finitely many colors, there is a monochromatic subordering $\mathbb Q$ isomorphic to $\mathbb P$. Chain-Ramsey partial orderings stratify naturally into levels. We show that a countably infinite partial ordering with finite levels is chain-Ramsey if and only if it is biembeddable with one of a canonical collection of examples constructed from certain edge-Ramsey families of finite bipartite graphs. A similar analysis applies to a large class of countably infinite partial orderings with infinite levels.

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Hindman’s theorem for sums along the full binary tree, $$\Sigma ^0_2$$-induction and the Pigeonhole principle for trees

Carlucci

Tavernelli²

2022

Arch. Math. Logic

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We formulate a restriction of Hindman’s Finite Sums Theorem in which monochromaticity is required only for sums corresponding to rooted finite paths in the full binary tree. We show that the resulting principle is equivalent to $$\Sigma ^0_2$$ Σ 2 0 -induction over $$\mathsf {RCA}_0$$ RCA 0 . The proof uses the equivalence of this Hindman-type theorem with the Pigeonhole Principle for trees $${\mathsf {T}\,}{\mathsf {T}}^1$$ T T 1 with an extra condition on the solution tree.

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Reverse mathematics and Ramsey's property for trees

Cited by 19 publications

References 4 publications

Coloring the rationals in reverse mathematics

Coloring the rationals in reverse mathematics

Ramsey Properties of Countably Infinite Partial Orderings

Hindman’s theorem for sums along the full binary tree, $$\Sigma ^0_2$$-induction and the Pigeonhole principle for trees

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