Ramsey’s Theorem on Trees Wei Li Joint Work with C. T. Chong, Wei Wang and Yue Yang matliw@nus.edu.sg Department of Mathematics, NUS Computability Theory and Foundations of Mathematics, Tokyo 20 September, 2016 1 / 18
Reverse Mathematics and Induction 1 Ramsey’s Theorem and Ramsey’s Theorem on Trees 2 TT 1 3 References 4 2 / 18
Reverse Mathematics and Induction Reverse Mathematics Main Question of Reverse Mathematics: What are the appropriate axioms for mathematics? History: 1970’s, Harvey Friedman and Stephen Simpson. Standard Reference: Subsystems of Second Order Arithmetic, by Simpson. 3 / 18
Reverse Mathematics and Induction Reverse Mathematics Language: The Language of Second Order Arithmetic. Model: �M , S � is a model of Second Order Arithmetic. M is a model of First Order Arithmetic. We use ω to denote the standard model of arithmetic. M may not be standard. S ⊆ P ( M ) . Axioms: Usual axioms of Peano Arithmetic (PA), where the induction is restricted to Σ 0 1 formulas Set Existence Axioms. 4 / 18
Reverse Mathematics and Induction Inductions in Reverse Mathematics Big Five: RCA 0 ⇐ WKL 0 ⇐ ACA 0 ⇐ ATR 0 ⇐ Π 1 1 -CA 0 WKL 0 ↾ First Order = Σ 0 1 Induction; ACA 0 ↾ First Order = PA. Induction: ∀ x ( ∀ y < x φ ( y ) ⇒ φ ( x )) ⇒ ∀ x ( φ ( x )) If φ is restricted to Σ 0 n formulas, then the induction is called Σ 0 n Induction (Denoted as I Σ 0 n , or I Σ n for short.) Similarly, we have I Π n , I ∆ n . Main Question on Induction: What are the appropriate inductions for mathematics? 5 / 18
Reverse Mathematics and Induction Inductions Axioms Bounding: ∀ y < x ( ∃ w φ ( y , w )) ⇒ ∃ b ( ∀ y < x ∃ w < b φ ( y , w )) If φ is restricted to Σ 0 n formulas, then the bounding is called Σ 0 n Bounding (Denoted as B Σ 0 n , or B Σ n for short.) Similarly, we have B Π n , B ∆ n . Theorem (Kirby and Paris) I Σ n ⇔ I Π n B Π n ⇔ B ∆ n + 1 ⇔ B Σ n + 1 I Σ n ⇒ B Σ n , B Σ n + 1 ⇒ I Σ n , B Σ n �⇒ I Σ n 6 / 18
Ramsey’s Theorem and Ramsey’s Theorem on Trees Ramsey’s Theorem X , H ⊆ M . Let [ X ] n be the collection of all subsets of X of size n . Coloring C : [ M ] n → k . Homogenous set H : C ↾ [ H ] n is a constant function. Theorem (Ramsey) Suppose k , n ≥ 1 . Every coloring C : [ M ] n → k has an infinite homogenous set. Notation: k , n are fixed. RT n k . n is fixed. RT n = ∀ k RT n k . 7 / 18
Ramsey’s Theorem and Ramsey’s Theorem on Trees Ramsey’s Theorem on Trees 2 < m : Collection of all ( M -finite) binary strings of length < m . 2 < M : Collection of all ( M -finite) binary strings in M . X , H ⊆ 2 < M . Let [ X ] n be the collection of all compatible subsets of X of size n . 2 < M � n → k . � Coloring C : Homogenous/Monochromatic tree H : H ∼ = 2 < m (Order Isomorphic, m ∈ M � {M} ) and C ↾ [ H ] n is a constant function. Theorem 2 < M � n → k has an infinite � Suppose k , n ≥ 1 . Every coloring C : monochromatic tree. 8 / 18
Ramsey’s Theorem and Ramsey’s Theorem on Trees Ramsey’s Theorem on Trees Notation: k , n are fixed. TT n k . n is fixed. TT n = ∀ k TT n k . TT n k ⇒ RT n k 9 / 18
Ramsey’s Theorem and Ramsey’s Theorem on Trees TT v.s. RT Theorem (Logicians) Axiom First Order Second Order (Over RCA 0 ) TT 1 > B Σ 2 , ≤ I Σ 2 > RCA 0 + B Σ 2 , ⊥ WKL 0 , < ACA 0 RT 1 RCA 0 + B Σ 2 B Σ 2 TT 2 > RT 2 ≥ B Σ 2 , ≤ I Σ 3 2 , < ACA 0 2 RT 2 ≥ B Σ 2 , < I Σ 2 > RCA 0 + B Σ 2 , ⊥ WKL 0 , < ACA 0 2 TT n k , n ≥ 3 , k ≥ 2 PA ACA 0 RT n PA ACA 0 k 10 / 18
TT 1 TT 1 Assuming I Σ 2 TT 1 ⇒ RT 1 ⇒ B Σ 2 . I Σ 2 ⇒ TT 1 2 < M � → k . � Given C : Consider the maximal c 0 < k such that ∃ σ ∀ τ ⊇ σ ( C ( τ ) ≥ c 0 ) . σ 0 is a witness for the c 0 . c 0 is dense among extensions of σ 0 . The monochromatic tree is recursive. 11 / 18
TT 1 Question 2 < M � → k . � Assume B Σ 2 + ¬ I Σ 2 and C : Is there an infinite monochromatic tree? What is the complexity of an monochromatic tree? Is there a monochromatic tree preserving B Σ 2 ? 12 / 18
TT 1 Density Theorem (Corduan, Groszek and Mileti) Suppose M | = B Σ 2 + ¬ I Σ 2 . There is k ∈ M with a recursive 2 < M � → k such that there is no recursive monochromatic tree. � C : Corollary RCA 0 + B Σ 2 �⊢ TT 1 . In that coloring C , every color is nowhere dense. 13 / 18
TT 1 Lowness Theorem (Chong, Li, Wang and Yang) Suppose M | = B Σ 2 + ¬ I Σ 2 . There is k ∈ M with a recursive 2 < M � → k such that there is no 0 ′ -recursive monochromatic tree. � C : Corollary WKL 0 + B Σ 2 �⊢ TT 1 . In that coloring C , no monochromatic tree is low. 14 / 18
TT 1 Existence Theorem (Chong, Li, Wang and Yang) 2 < M � → k is recursive. There is a Suppose M | = B Σ 2 + ¬ I Σ 2 and C : � regular monochromatic tree. A set X is regular, if X ∩ M -finite = M -finite. Non-definable solution. 15 / 18
TT 1 Complexity Theorem (Chong, Li, Wang and Yang) Suppose M | = B Σ 2 + ¬ I Σ 2 . 2 < M � → k such that there is no � There is k ∈ M with a recursive C : recursive monochromatic tree but there is a low monochromatic tree. 2 < M � → k such that there is � There is k ∈ M with a recursive C : no low monochromatic tree but there is a monochromatic tree preserving B Σ 2 . Conjecture TT 1 �⊢ I Σ 2 . 16 / 18
References References C. T. Chong, Theodore A. Slaman and Yue Yang. The inductive strength of Ramsey’s theorem for pairs. To appear. Jennifer Chubb, Jeffry L. Hirst, and Timothy H. McNicholl, Reverse mathematics, computability, and partitions of trees, Journal of Symbolic Logic, vol. 74 (2009), no. 1, pp. 201–215. Jared Corduan, Marcia J. Groszek and Joseph R. Mileti, Reverse mathematics and Ramsey’s property for trees, Journal of Symbolic Logic, vol. 75 (2010), no. 3, pp. 945–954. Damir Dzhafarov and Ludovic Patey, Coloring trees in reverse mathematics. To appear. Stephen G. Simpson. Subsystems of Second Order Arithmetric. Heidelberg, Springer–Verlag, 1999. 17 / 18
Thank you. 18 / 18
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