An effective version of Hall’s theorem

Kierstead, H. A.

doi:10.1090/s0002-9939-1983-0691291-0

Cited by 7 publications

(13 citation statements)

References 6 publications

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“…The next lemma means that it is actually equivalent to

sans-serifWKL

over

{RCA}_{0}

. Lemma

{sans-serifRCA}_{sans-serif0} ⊢ [B^{''}, G^{''}, {normalH}^{'}] \to WKL

, i.e., the following assertion implies

sans-serifWKL

over

{RCA}_{0}

: If

R (B, G)

is a bipartite graph which is

B, G

‐highly recursive and satisfies the expanding Hall condition, then

R (B, G)

has a solution. Proof We extend Kierstead's proof of [, Theorem 5] to show this lemma. We reason in

{RCA}_{0}

.…”

Section: Marriage Theorems With Expanding and Recursive Expanding Halmentioning

confidence: 94%

“…Kierstead [, Theorem 5] showed that

[{normalB}^{''}, {normalG}^{''}, H^{'}]

does not hold in the least ω‐model of

{RCA}_{0}

, while Hirst [, Theorem 2.3] showed that it is provable in

{WKL}_{0}

. The next lemma means that it is actually equivalent to

sans-serifWKL

over

{RCA}_{0}

.…”

Section: Marriage Theorems With Expanding and Recursive Expanding Halmentioning

confidence: 99%

“…However, in the early age of recursive graph theory, cf., e.g., , Manaster and Rosenstein found that such a multi‐valued function need not have a single‐valued computable injective selection, even if it has a computable graph and its local finiteness is computably confirmed. To render the marriage theorem computable , Kierstead introduced the notion of expanding Hall condition, which indicates that the difference between

| R [X] |

and

| X |

tends to infinity as

| X |

tends to infinity, where X ranges over all finite subsets of B . Then, he found that R has a single‐valued computable injective selection whenever the graph of R is computable , the local finiteness of R is computably confirmed and the expanding Hall condition for R is computably witnessed.…”

Section: Introductionmentioning

confidence: 99%

“…Using this terminology, we can rephrase the result of Hirst as follows: The statement [B′, G, H] is equivalent to

sans-serifACA

, and [

{normalB}^{''}, G, H

] is equivalent to

sans-serifWKL

over

{RCA}_{0}

. Furthermore, Kierstead showed that [

{normalB}^{''}, {normalG}^{''}, {normalH}^{''}

] holds effectively. Now we note that the Hall condition has

Π_{2}^{0}

form but it can be written as a

Π_{1}^{0}

formula under the assumption of being B ‐highly recursive.…”

Section: Introductionmentioning

confidence: 99%

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On the strength of marriage theorems and uniformity

Fujiwara

Higuchi

Kihara

2014

Mathematical Logic Qtrly

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Kierstead showed that every computable marriage problem has a computable matching under the assumption of computable expanding Hall condition and computable local finiteness for boys and girls. The strength of the marriage theorem reaches WKL 0 or ACA 0 if computable expanding Hall condition or computable local finiteness for girls is weakened. In contrast, the provability of the marriage theorem is maintained in RCA even if local finiteness for boys is completely removed. Using these conditions, we classify the strength of variants of marriage theorems in the context of reverse mathematics. Furthermore, we introduce another condition that also makes the marriage theorem provable in RCA 0 , and investigate the sequential and Weihrauch strength of marriage theorems under that condition.

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“…The next lemma means that it is actually equivalent to

sans-serifWKL

over

{RCA}_{0}

. Lemma

{sans-serifRCA}_{sans-serif0} ⊢ [B^{''}, G^{''}, {normalH}^{'}] \to WKL

, i.e., the following assertion implies

sans-serifWKL

over

{RCA}_{0}

: If

R (B, G)

is a bipartite graph which is

B, G

‐highly recursive and satisfies the expanding Hall condition, then

R (B, G)

has a solution. Proof We extend Kierstead's proof of [, Theorem 5] to show this lemma. We reason in

{RCA}_{0}

.…”

Section: Marriage Theorems With Expanding and Recursive Expanding Halmentioning

confidence: 94%

“…Kierstead [, Theorem 5] showed that

[{normalB}^{''}, {normalG}^{''}, H^{'}]

does not hold in the least ω‐model of

{RCA}_{0}

, while Hirst [, Theorem 2.3] showed that it is provable in

{WKL}_{0}

. The next lemma means that it is actually equivalent to

sans-serifWKL

over

{RCA}_{0}

.…”

Section: Marriage Theorems With Expanding and Recursive Expanding Halmentioning

confidence: 99%

| R [X] |

and

| X |

tends to infinity as

| X |

Section: Introductionmentioning

confidence: 99%

“…Using this terminology, we can rephrase the result of Hirst as follows: The statement [B′, G, H] is equivalent to

sans-serifACA

, and [

{normalB}^{''}, G, H

] is equivalent to

sans-serifWKL

over

{RCA}_{0}

. Furthermore, Kierstead showed that [

{normalB}^{''}, {normalG}^{''}, {normalH}^{''}

] holds effectively. Now we note that the Hall condition has

Π_{2}^{0}

form but it can be written as a

Π_{1}^{0}

formula under the assumption of being B ‐highly recursive.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On the strength of marriage theorems and uniformity

Fujiwara

Higuchi

Kihara

2014

Mathematical Logic Qtrly

View full text Add to dashboard Cite

show abstract

“…To find a matching containing a given set of vertices, we have the following interesting result. Proposition 1.3.2 (Kierstead [316]) Let G be a graph and T the set of vertices of degree ∆(G). If G[T ] is a bipartite subgraph, then there exists a matching in G saturating all vertices of T .…”

Section: Matchings In Non-bipartite Graphsmentioning

confidence: 99%