Recursive Marriage Theorems and Reverse Mathematics - Google Sites

Recursive Marriage Theorems and Reverse Mathematics Makoto Fujiwara1 Tohoku University

2012.2.21

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Joint work with Kojiro Higuchi and Takayuki Kihara

Makoto Fujiwara (Tohoku University)

Recursive Marriage Theorems and Reverse Mathematics

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Contents

1

What is Reverse Mathematics?

2

Previous Study on the Strength of Marriage Theorems.

3

Marriage Theorems with a lot of acquaintances.

4

Marriage Theorems with a few acquaintances.



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What is Reverse Mathematics? The aim of reverse mathematics is classifying the mathematical theorems by the difficulty. We look for the set existence axiom which is exactly needed to prove each theorem. Actually, we check which set existence axiom is necessary and sufficient to the theorem over base theory RCA0 . Here RCA0 consists of the following axioms. 1 2 3

Basic axioms of arithmetic. Σ01 induction axiom. ∆01 (recursive) set existence axiom.

In reverse mathematics, we prove not only a theorem from axioms but also an axiom from the theorem. Then this research program is called “reverse mathematics”. . Makoto Fujiwara (Tohoku University)


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What is Reverse Mathematics? Although there are so many mathematical theorems, most of ordinary mathematical theorems are provable in RCA0 or equivalent to WKL, ACA, ATR, or Π11 -CA over RCA0 . RCA0 (+some strong induction axiom) corresponds to recursive mathematics and the equivalence to WKL, ACA... reveals how far is the theorem from holding recursively. RCA0 < WKL0 < ACA0 < ATR0 < Π11 -CA0 Here WKL0 is the theory consisting of RCA0 + WKL and so on. In my talk, let graphs be countable unless otherwise stated. Makoto Fujiwara (Tohoku University)


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Reverse Mathematics for Graph Theory Theorem in Graph Theory Π11 -CA0 ATR0 ACA0

WKL0

RCA0

For a sequence of graphs, there exists a function chosing the graphs which has a Hamilton path. For a sequence of graphs having at most one Ham. path, there exists a function chosing the graphs which has a Ham. path. ¨ Konig’s lemma. Marriage theorem and symmetric marriage theorem. Every graph can be decomposed into connected components. Ramsey theorem for k size and 2 coloring. (k ≥ 3) Marriage theorem and symmetric marriage theorem for highly recursive graphs. Every k-colorable highly recursive graph is (2k − 2)-colorable. Every k-colorable graph is m-colorable. (m ≥ k ≥ 2) Theorems for finite graphs. Every k-colorable highly recursive graph is (2k − 1)-colorable.



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Contents

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3


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B′H GM(Infinite Marriage Theorem) (M.Hall, 1948) A bipartite graph (B, G; R) which satisfies condition H and B′ , has a solution. Condition H : For all X⊂fin B,

|X| ≤ |N(X)| (= |{g | ∃b ∈ X ((b, g) ∈ R)}| ) holds. Condition B′ : For all b ∈ B, |N(b)| < ∞ holds. A solution : A injection f ⊂ R from B to G

B

G

.. . . ..



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B

G

.. . . ..



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B

G

.. . . ..



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B

G

. ..

Figure: BH GM does not hold. (Condition B′ can not be dropped.) Makoto Fujiwara (Tohoku University)


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Theorem. (Manaster-Rosenstein, 1972) There exists a recursive bipartite graph (B, G; R) which satisfies condition H and B′ , but has no recursive solution. (B′H GM does not hold recursively.)

b

→

N is recursive. ∈

∈

Condition B′′ : The function p : B

7→ |N(b)|

g

∈

∈

Condition G′ : For all g ∈ G, |N(g)| < ∞ holds. Condition G′′ : The function p : G → N is recursive.

7→ |N(g)|

Theorem. (Manaster-Rosenstein, 1972) There exists a recursive bipartite graph (B, G; R) which satisfies condition H, B′′ and G′′ , but has no recursive solution. (B′′H G′′ M does not hold recursively.) Makoto Fujiwara (Tohoku University)


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b

→

N is recursive. ∈

∈


7→ |N(b)|

g

∈

∈


7→ |N(g)|



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b

→

N is recursive. ∈

∈


7→ |N(b)|

g

∈

∈


7→ |N(g)|



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Theorem (Hirst, 1990) 1 2

RCA0 ⊢ B′H GM ↔ ACA RCA0 ⊢ B′′ GM ↔ WKL H

B′′H GM :

A bipartite graph (B, G; R) which satisfies condition H and B′′ , has a solution.

Note that condition B′′ is written as follows within RCA0 .

∃p : B → N such that ∀b, g((b, g) ∈ R → g ≤ p(b))



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Theorem (Hirst, 1990) 1 2

RCA0 ⊢ B′H GM ↔ ACA RCA0 ⊢ B′′ GM ↔ WKL H

B′′H GM :

A bipartite graph (B, G; R) which satisfies condition H and B′′ , has a solution.

Note that condition B′′ is written as follows within RCA0 .

∃p : B → N such that ∀b, g((b, g) ∈ R → g ≤ p(b))



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B′H G′H Ms (Symmetric Marriage Theorem) A bipartite graph (B, G; R) which satisfies condition H for B and G, B′ and G′ , has a symmetric solution (bijection f ⊂ R from B to G). Theorem (Hirst, 1990) 1 2

RCA0 ⊢ B′H G′H Ms ↔ ACA RCA0 ⊢ B′′ G′′ M ↔ WKL H H s

B′′H G′′H M :

A bipartite graph (B, G; R) which satisfies condition H for B and G, B′′ and G′′ , has a symmetric solution.



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B′H G′H Ms (Symmetric Marriage Theorem) A bipartite graph (B, G; R) which satisfies condition H for B and G, B′ and G′ , has a symmetric solution (bijection f ⊂ R from B to G). Theorem (Hirst, 1990) 1 2

RCA0 ⊢ B′H G′H Ms ↔ ACA RCA0 ⊢ B′′ G′′ M ↔ WKL H H s

B′′H G′′H M :

A bipartite graph (B, G; R) which satisfies condition H for B and G, B′′ and G′′ , has a symmetric solution.



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Contents

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Theorem. (Manaster-Rosenstein, 1972 (Review)) B′′H G′′ M does not hold recursively. Condition H′ : Condition H and

∀n∃m∀X⊂fin B (|X| ≥ m → |N(X)| − |X| ≥ n) holds. Condition H′′ : Condition H and there exists a recursive function h : N → N s.t.

∀n∀X⊂fin B (|X| ≥ h(n) → |N(X)| − |X| ≥ n) holds. Theorem. (Kierstead, 1983) B′′H′′ G′′ M holds recursively. Theorem. (Kierstead, 1983) B′′H′ G′′ M does not hold recursively. Makoto Fujiwara (Tohoku University)


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Theorem. (Manaster-Rosenstein, 1972 (Review)) B′′H G′′ M does not hold recursively. Condition H′ : Condition H and

∀n∃m∀X⊂fin B (|X| ≥ m → |N(X)| − |X| ≥ n) holds. Condition H′′ : Condition H and there exists a recursive function h : N → N s.t.

∀n∀X⊂fin B (|X| ≥ h(n) → |N(X)| − |X| ≥ n) holds. Theorem. (Kierstead, 1983) B′′H′′ G′′ M holds recursively. Theorem. (Kierstead, 1983) B′′H′ G′′ M does not hold recursively. Makoto Fujiwara (Tohoku University)


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Facts (Review) B′H GM holds. (Infinite marriage theorem) BH GM does not hold. Proposition. (Kierstead, 1983)) A bipartite graph (B, G; R) which satisfies condition H′ , has a solution. That is, BH′ GM holds.



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The Strength of Marriage Theorems

H

H′

H′′

BH GM BH G′ M BH G′′ M

BH′ GM BH′ G′ M BH′ G′′ M

BH′′ GM BH′′ G′ M BH′′ G′′ M

Hirst

B′H GM B′H G′ M B′H G′′ M

B′H′ GM B′H′ G′ M B′H′ G′′ M

B′H′′ GM B′H′′ G′ M B′H′′ G′′ M

Hirst

B′′H GM B′′H G′ M B′′H G′′ M

B′′H′ GM B′′H′ G′ M B′′H′ G′′ M (nonrecursive)

B′′H′′ GM B′′H′′ G′ M B′′H′′ G′′ M (recursive)

False

ACA ← −−→

WKL ← −−→

H′′ : ∃h : N → N (h(0) = 0 ∧ ∀n∀X⊂fin B (|X| ≥ h(n) → |N(X)| − |X| ≥ n)) Makoto Fujiwara (Tohoku University)


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H ∗ ∗ ∗

H′

H′′


BH′′ GM BH′′ G′ M BH′′ G′′ M B′H′′ GM B′H′′ G′ M B′H′′ G′′ M

Hirst



Hirst


B′′H′ GM B′′H′′ GM ′′ ′ BH′ G M B′′H′′ G′ M B′′H′ G′′ M B′′H′′ G′′ M ⊣ RCA0 (nonrecursive)

ACA ← −−→

WKL ← −−→



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H ∗ ∗ ∗

H′

H′′



Hirst



B′H′′ GM B′H′′ G′ M B′H′′ G′′ M

Hirst


B′′H′ GM B′′H′ G′ M B′′H′ G′′ M

B′′H′′ GM B′′H′′ G′ M B′′H′′ G′′ M ⊣ RCA0

ACA ← −−→

WKL ← −−→



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H ∗ ∗ ∗ Hirst

H′

H′′



ACA ← −−→



B′H′′ GM B′H′′ G′ M B′H′′ G′′ M

WKL


B′′H′ GM B′′H′ G′ M B′′H′ G′′ M




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H ∗ ∗ ∗ Hirst

H′

H′′



ACA ← −−→



B′H′′ GM B′H′′ G′ M B′H′′ G′′ M

WKL


B′′H′ GM B′′H′ G′ M B′′H′ G′′ M




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H ∗ ∗ ∗ Hirst

H′

H′′



ACA ← −−→



B′H′′ GM B′H′′ G′ M B′H′′ G′′ M

WKL


B′′H′ GM B′′H′ G′ M B′′H′ G′′ M




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H ∗ ∗ ∗

H′

H′′



ACA



B′H′′ GM B′H′′ G′ M B′H′′ G′′ M

WKL


B′′H′ GM B′′H′ G′ M B′′H′ G′′ M




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H ∗ ∗ ∗

H′

H′′


BH′′ GM BH′′ G′ M BH′′ G′′ M ⊣ RCA0 + Σ03 -IND

ACA



B′H′′ GM B′H′′ G′ M B′H′′ G′′ M

WKL


B′′H′ GM B′′H′ G′ M B′′H′ G′′ M




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H ∗ ∗ ∗

H′

H′′


BH′′ GM BH′′ G′ M BH′′ G′′ M ⊣ RCA0 + Σ03 -IND

ACA



B′H′′ GM B′H′′ G′ M B′H′′ G′′ M ⊣ RCA0 + Σ03 -IND

WKL


B′′H′ GM B′′H′ G′ M B′′H′ G′′ M




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The Strength of Marriage Theorems Corollary. (RCA0 + Σ03 -IND ⊢ BH′′ G′′ M) BH′′ G′′ M holds recursively. Corollary. (RCA0 ⊢ B′′H′′ G′ M → WKL) B′′H′′ G′ M does not hold recursively. Theorem. (Kierstead, Review) B′′H′ G′′ M does not hold recursively. Remark.

Condition H′′ and G′′ are essential to make marriage theorem hold recursively. Makoto Fujiwara (Tohoku University)


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The Strength of Marriage Theorems Corollary. (RCA0 + Σ03 -IND ⊢ BH′′ G′′ M) BH′′ G′′ M holds recursively. Corollary. (RCA0 ⊢ B′′H′′ G′ M → WKL) B′′H′′ G′ M does not hold recursively. Theorem. (Kierstead, Review) B′′H′ G′′ M does not hold recursively. Remark.

Condition H′′ and G′′ are essential to make marriage theorem hold recursively. Makoto Fujiwara (Tohoku University)


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On the Strength of Symmetric Marriage Theorems

BH GH Ms BH G′H Ms BH G′′ M H s B′H GH Ms B′H G′H Ms (Hirst) B′H G′′ M H s B′′ G M H H s ′′ ′ BH GH Ms B′′ G′′ M (Hirst) H H s

BH GH′ Ms BH G′H′ Ms BH G′′ M H′ s B′H GH′ Ms B′H G′H′ Ms B′H G′′ M H′ s ′ B′′ G Ms H H ′′ ′ BH GH′ Ms B′′ G′′ M H H′ s False


BH GH′′ Ms BH G′H′′ Ms BH G′′ M H′′ s B′H GH′′ Ms B′H G′H′′ Ms B′H G′′ M H′′ s ′′ B′′ G Ms H H ′′ ′ BH GH′′ Ms B′′ G′′ M H H′′ s ACA0

BH′ GH′ Ms BH′ G′H′ Ms BH′ G′′ M H′ s B′H′ GH′ Ms B′H′ G′H′ Ms B′H′ G′′ M H′ s ′ B′′ G Ms H′ H ′′ ′ BH′ GH′ Ms G′′ M B′′ H′ H′ s

WKL0

BH′ GH′′ Ms BH′ G′H′′ Ms BH′ G′′ M H′′ s B′H′ GH′′ Ms B′H′ G′H′′ Ms B′H′ G′′ M H′′ s ′′ B′′ G Ms H′ H ′′ ′ BH′ GH′′ Ms G′′ M B′′ H′ H′′ s

BH′′ GH′′ Ms BH′′ G′H′′ Ms BH′′ G′′ M H′′ s B′H′′ GH′′ Ms B′H′′ G′H′′ Ms B′H′′ G′′ M H′′ s ′′ B′′ G Ms H′′ H ′′ ′ BH′′ GH′′ Ms G′′ M B′′ H′′ H′′ s

RCA0


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Contents

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We consider a marriage theorem such that B = G = ω. IM (Recasting of Infinite Marriage Theorem) A bipartite graph (ωB , ωG ; R) which satisfies condition H and ∀i, j∃t((i, j) ∈ R → j ≤ i + t), has a solution. There exists a recursive R such that G = (ωB , ωG ; R) satisfies condition H and there exists a recursive q : ω → ω s.t. ∀i, j((i, j) ∈ R → j ≤ i + q(i)) holds, but does not have a recursive solution. (RCA0 ⊢ B′′H GM ↔ WKL (Hirst).) If q ≡ 0, {(i, i) | i ∈ ω} is a solution of G, namely, G has a recursive solution.

Where is the border?



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We consider a marriage theorem such that B = G = ω. IM (Recasting of Infinite Marriage Theorem) A bipartite graph (ωB , ωG ; R) which satisfies condition H and ∀i, j∃t((i, j) ∈ R → j ≤ i + t), has a solution. There exists a recursive R such that G = (ωB , ωG ; R) satisfies condition H and there exists a recursive q : ω → ω s.t. ∀i, j((i, j) ∈ R → j ≤ i + q(i)) holds, but does not have a recursive solution. (RCA0 ⊢ B′′H GM ↔ WKL (Hirst).) If q ≡ 0, {(i, i) | i ∈ ω} is a solution of G, namely, G has a recursive solution. We see q : N → N as a parameter, and investigate the strength of the following assertion IM(q) with respect to reverse mathematics. IM(q) :

A bipartite graph (NB , NG ; R) which satisfies condition H and ∀i, j((i, j) ∈ R → j ≤ i + q(i)), has a solution.



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Theorem. 1

RCA0 ⊢ q is bounded by standard constant → IM(q).

2

RCA0 + Π02 -IND ⊢ q is bounded → IM(q).

3

RCA0 ⊢ q is unbounded and nondecreasing → (IM(q) ↔ WKL).

IM(q) :


Corollary. For any recursive R ⊂ ω × ω and q : ω → ω which satisfies condition H and ∀i, j ((i, j) ∈ R → j ≤ i + q(i)),

G = (ωB , ωG ; R) has a recursive solution if q is bounded. G does not necessary have a recursive solution if q is unbounded. Makoto Fujiwara (Tohoku University)


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Theorem. 1

RCA0 ⊢ q is bounded by standard constant → IM(q).

2

RCA0 + Π02 -IND ⊢ q is bounded → IM(q).

3

RCA0 ⊢ q is unbounded and nondecreasing → (IM(q) ↔ WKL).

IM(q) :


Corollary. For any recursive R ⊂ ω × ω and q : ω → ω which satisfies condition H and ∀i, j ((i, j) ∈ R → j ≤ i + q(i)),

G = (ωB , ωG ; R) has a recursive solution if q is bounded. G does not necessary have a recursive solution if q is unbounded. Makoto Fujiwara (Tohoku University)


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Theorem. (Recasting of RCA0 + Π02 -IND ⊢ q is bounded → IM(q)) RCA0 + Π02 -IND ⊢ IM∗ . IM∗ :

A bipartite graph (NB , NG ; R) which satisfies condition H and ∃k∀i, j((i, j) ∈ R → j ≤ i + k), has a solution.

WKL0 ⊢ IM∗ and RCA0 + Π02 -IND ⊢ IM∗ both hold. Question. RCA0 0 IM∗ ? (RCA0 ⊢ IM∗ ↔ WKL ∨ Π02 -IND ?)



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Theorem. (Recasting of RCA0 + Π02 -IND ⊢ q is bounded → IM(q)) RCA0 + Π02 -IND ⊢ IM∗ . IM∗ :

A bipartite graph (NB , NG ; R) which satisfies condition H and ∃k∀i, j((i, j) ∈ R → j ≤ i + k), has a solution.

WKL0 ⊢ IM∗ and RCA0 + Π02 -IND ⊢ IM∗ both hold. Question. RCA0 0 IM∗ ? (RCA0 ⊢ IM∗ ↔ WKL ∨ Π02 -IND ?)



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Definition. Condition H+ : ∀X⊂fin B∃k (|X| ≤ |N(X)| ≤ |X| + k) Condition H++ : ∃k∀X⊂fin B (|X| ≤ |N(X)| ≤ |X| + k) Condition H+ includes condition B′ . Proposition. (Recasting of B′H GM) A bipartite graph (B, G; R) which satisfies condition H+ , has a solution. (B′H+ GM holds.) Corollary. (RCA0 + Π02 -IND ⊢ IM∗ ) RCA0 + Π02 -IND ⊢ B′′H++ GM (A bipartite graph (B, G; R) which satisfies condition H++ and B′′ , has a solution).



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Definition. Condition H+ : ∀X⊂fin B∃k (|X| ≤ |N(X)| ≤ |X| + k) Condition H++ : ∃k∀X⊂fin B (|X| ≤ |N(X)| ≤ |X| + k) Condition H+ includes condition B′ . Proposition. (Recasting of B′H GM) A bipartite graph (B, G; R) which satisfies condition H+ , has a solution. (B′H+ GM holds.) Corollary. (RCA0 + Π02 -IND ⊢ IM∗ ) RCA0 + Π02 -IND ⊢ B′′H++ GM (A bipartite graph (B, G; R) which satisfies condition H++ and B′′ , has a solution).



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Theorem. RCA0 ⊢ B′H++ GM (A bipartite graph (B, G; R) which satisfies k=1 condition H++ k=1 , has a solution). Conjecture RCA0 + X-IND ⊢ B′H++ GM (A bipartite graph (B, G; R) which satisfies condition H++ , has a solution).



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Theorem. A recursive bipartite graph (B, G; R) which satisfies condition H++ and G′ , has a recursive solution. That is, B′H++ G′ M holds recursively. The above theorem does not require the recursiveness of locally finiteness. This suggests that RCA0 + X-IND ⊢ B′H++ G′ M, but we have not checked it. On the other hand, the followings hold. Proposition. 1

RCA0 ⊢ B′H+ G′ M ↔ ACA.

2

RCA0 ⊢ B′′2H+ G′′ M ↔ WKL.

Condition 2H+ : ∀X⊂fin B (|X| ≤ |N(X)| ≤ 2|X|) Makoto Fujiwara (Tohoku University)


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Theorem. A recursive bipartite graph (B, G; R) which satisfies condition H++ and G′ , has a recursive solution. That is, B′H++ G′ M holds recursively. The above theorem does not require the recursiveness of locally finiteness. This suggests that RCA0 + X-IND ⊢ B′H++ G′ M, but we have not checked it. On the other hand, the followings hold. Proposition. 1

RCA0 ⊢ B′H+ G′ M ↔ ACA.

2

RCA0 ⊢ B′′2H+ G′′ M ↔ WKL.

Condition 2H+ : ∀X⊂fin B (|X| ≤ |N(X)| ≤ 2|X|) Makoto Fujiwara (Tohoku University)


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Bibliography 1

W. Gasarch, “A survey of recursive combinatorics”, Handbook of recursive mathematics, Vol. 2, Stud. Logic Found. Math., 139, Amsterdam: North-Holland(1998), pp. 1041–1176.

2

J. R. Hirst, Marriage theorems and reverse mathematics, Contemporary Mathematics 106 (1990), pp. 181-196.

3

H. A. Kierstead, An effective version of Hall’s theorem, American Mathematical Society, 88 (1983), pp. 124–128.

4

A. Manaster and J. Rosenstein, Effective matchmaking (recursion theoretic aspects of a theorem of Philip Hall), Proc. Amer. Math. Soc.,25 (1972), pp. 615–654.

Thank you for your attention. Makoto Fujiwara (Tohoku University)


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