Colorful well-foundedness Yann Pequignot University of California, - PowerPoint PPT Presentation

Colorful well-foundedness Yann Pequignot University of California, Los Angeles YST workshop, Bernoulli Center, Lausanne June 29, 2018

Part I Well-quasi-orders and Better-quasi-orders

Well-quasi-orders A quasi-order (qo) is a set Q together with a reflexive and transitive binary relation ⩽ . Definition A well-quasi-order (wqo) is a qo that satisfies one of the following equivalent conditions. 1 Q is well-founded and has no infinite antichain;

Well-quasi-orders A quasi-order (qo) is a set Q together with a reflexive and transitive binary relation ⩽ . Definition A well-quasi-order (wqo) is a qo that satisfies one of the following equivalent conditions. 1 Q is well-founded and has no infinite antichain; 2 there exists no bad sequence, i.e. no ( q n ) n s.t. m < n → q m ⩽̹ q n . ∀ m , n ∈ ω

Well-quasi-orders A quasi-order (qo) is a set Q together with a reflexive and transitive binary relation ⩽ . Definition A well-quasi-order (wqo) is a qo that satisfies one of the following equivalent conditions. 1 Q is well-founded and has no infinite antichain; 2 there exists no bad sequence, i.e. no ( q n ) n s.t. m < n → q m ⩽̹ q n . ∀ m , n ∈ ω 3 P ( Q ) is well-founded, under: X ⩽ Y ← → ∀ x ∈ X ∃ y ∈ Y x ⩽ y .

Well-quasi-orders Examples of wqos Finite quasi-orders

Well-quasi-orders Examples of wqos Finite quasi-orders Well-orders

Well-quasi-orders Examples of wqos Finite quasi-orders Well-orders If P and Q are wqo, then P × Q is wqo.

Well-quasi-orders Examples of wqos Finite quasi-orders Well-orders If P and Q are wqo, then P × Q is wqo. (Higman 52’) If P is wqo, then P <ω is wqo under ( p i ) i < n ⩽ ( q j ) j < m ← → ∃ f : n → m strictly increasing s.t. p i ⩽ q f ( i ) for all i < n

Well-quasi-orders Examples of wqos Finite quasi-orders Well-orders If P and Q are wqo, then P × Q is wqo. (Higman 52’) If P is wqo, then P <ω is wqo under ( p i ) i < n ⩽ ( q j ) j < m ← → ∃ f : n → m strictly increasing s.t. p i ⩽ q f ( i ) for all i < n (Laver 71’) Countable linear orders under embeddability.

Well-quasi-orders Examples of wqos Finite quasi-orders Well-orders If P and Q are wqo, then P × Q is wqo. (Higman 52’) If P is wqo, then P <ω is wqo under ( p i ) i < n ⩽ ( q j ) j < m ← → ∃ f : n → m strictly increasing s.t. p i ⩽ q f ( i ) for all i < n (Laver 71’) Countable linear orders under embeddability. (Robertson-Seymour, 500 pages, 1983-2004) The finite undirected graphs under the minor relation.

A wqo Q such that P ( Q ) is not wqo . . . . . . . . . . . . . . . . . . . . . . . . . . . ... • • • • • • • • • · · · • • • • • • • • · · · • • • • • • • · · · • • • • • • · · · Richard Rado, 1954. • • • • • · · · • • • • · · · R is defined on [ ω ] 2 by: • • • · · · • • · · · { m 0 , m 1 } ⩽ { n 0 , n 1 } • · · · iff { m 0 = n 0 and m 1 ⩽ n 1 , or Rado’s poset R m 0 < m 1 < n 0 < n 1

A wqo Q such that P ( Q ) is not wqo . . . . . . . . . . . . . . . . . . . . . . . . . . . ... • • • • • • • • • · · · • • • • • • • • · · · • • • • • • • · · · • • • • • • · · · Richard Rado, 1954. • • • • • · · · • • • • · · · R is defined on [ ω ] 2 by: • • • · · · • • · · · { m 0 , m 1 } ⩽ { n 0 , n 1 } • · · · iff { m 0 = n 0 and m 1 ⩽ n 1 , or Rado’s poset R m 0 < m 1 < n 0 < n 1

A wqo Q such that P ( Q ) is not wqo . . . . . . . . . . . . . . . . . . . . . . . . . . . ... • • • • • • • • • · · · • • • • • • • • · · · • • • • • • • · · · • • • • • • · · · Richard Rado, 1954. • • • • • · · · • • • • · · · R is defined on [ ω ] 2 by: • • • · · · • • · · · { m 0 , m 1 } ⩽ { n 0 , n 1 } • · · · iff { m 0 = n 0 and m 1 ⩽ n 1 , or Rado’s poset R m 0 < m 1 < n 0 < n 1

A wqo Q such that P ( Q ) is not wqo . . . . . . . . . . . . . . . . . . . . . . . . . . . ... • • • • • • • • • · · · • • • • • • • • · · · • • • • • • • · · · • • • • • • · · · Richard Rado, 1954. • • • • • · · · • • • • · · · R is defined on [ ω ] 2 by: • • • · · · • • · · · { m 0 , m 1 } ⩽ { n 0 , n 1 } • · · · iff { m 0 = n 0 and m 1 ⩽ n 1 , or Rado’s poset R m 0 < m 1 < n 0 < n 1

A wqo Q such that P ( Q ) is not wqo . . . . . . . . . . . . . . . . . . . . . . . . . . . ... • • • • • • • • • · · · • • • • • • • • · · · • • • ⃝ • • • ⃝ • · · · • • • • • • · · · Richard Rado, 1954. • • • • • · · · • • • • · · · R is defined on [ ω ] 2 by: • • • · · · • • · · · { m 0 , m 1 } ⩽ { n 0 , n 1 } • · · · iff { m 0 = n 0 and m 1 ⩽ n 1 , or m 0 < m 1 < n 0 < n 1

Better quasi-orders Fix a quasi-order Q and treat the element of Q as atoms , namely they have no elements but they are different from the empty set. We define by transfinite recursion: Q ∗ 0 = Q Q ∗ α + 1 = P ∗ ( Q ∗ (the non-empty subsets of V ∗ α ) α ) Q ∗ ∪ Q ∗ λ = α , for λ limit. α<λ Let Q ∗ = ∪ Q ∗ α . α We define a quasi-order on Q ∗ via the existence of a winning strategy in a suitable game G ( X , Y ) . Definition (Intuitive definition) A quasi-order Q is a better-quasi-order if Q ∗ is well-founded.

Making sense of the definition Let ( X n ) n be a sequence in Q ∗ such that m < n implies X n ⩽̹ X m .

Making sense of the definition Let ( X n ) n be a sequence in Q ∗ such that m < n implies X n ⩽̹ X m . For every infinite subset N = { n 0 , n 1 , n 2 , . . . } of ω , contemplate: G ( X n 0 , X n 1 ) G ( X n 1 , X n 2 ) G ( X n 2 , X n 3 ) G ( X n 3 , X n 4 ) G ( X n 4 , X n 5 )                                              σ n 0 , n 1 σ n 1 , n 2 σ n 2 , n 3 σ n 3 , n 4 σ n 4 , n 5 I II I II I II I II I II

Making sense of the definition Let ( X n ) n be a sequence in Q ∗ such that m < n implies X n ⩽̹ X m . For every infinite subset N = { n 0 , n 1 , n 2 , . . . } of ω , contemplate: G ( X n 0 , X n 1 ) G ( X n 1 , X n 2 ) G ( X n 2 , X n 3 ) G ( X n 3 , X n 4 ) G ( X n 4 , X n 5 )                                              σ n 0 , n 1 I II I II I II I II I II σ n 1 , n 2 σ n 2 , n 3 σ n 3 , n 4 σ n 4 , n 5 Y 0 0 Where S ( N ) = N ∖ { min N } . This defines a map f : [ ω ] ∞ → Q , f ( N ) = Q N , such that f ( N ) ⩽̹ f ( S ( N )) .

Making sense of the definition Let ( X n ) n be a sequence in Q ∗ such that m < n implies X n ⩽̹ X m . For every infinite subset N = { n 0 , n 1 , n 2 , . . . } of ω , contemplate: G ( X n 0 , X n 1 ) G ( X n 1 , X n 2 ) G ( X n 2 , X n 3 ) G ( X n 3 , X n 4 ) G ( X n 4 , X n 5 )                                              σ n 0 , n 1 σ n 1 , n 2 I II I II I II I II I II σ n 2 , n 3 σ n 3 , n 4 σ n 4 , n 5 Y 0 Y 0 0 1 Where S ( N ) = N ∖ { min N } . This defines a map f : [ ω ] ∞ → Q , f ( N ) = Q N , such that f ( N ) ⩽̹ f ( S ( N )) .

Making sense of the definition Let ( X n ) n be a sequence in Q ∗ such that m < n implies X n ⩽̹ X m . For every infinite subset N = { n 0 , n 1 , n 2 , . . . } of ω , contemplate: G ( X n 0 , X n 1 ) G ( X n 1 , X n 2 ) G ( X n 2 , X n 3 ) G ( X n 3 , X n 4 ) G ( X n 4 , X n 5 )                                              σ n 0 , n 1 σ n 1 , n 2 I II I II I II I II I II σ n 2 , n 3 σ n 3 , n 4 σ n 4 , n 5 Y 0 Y 0 Y 0 0 1 1 Where S ( N ) = N ∖ { min N } . This defines a map f : [ ω ] ∞ → Q , f ( N ) = Q N , such that f ( N ) ⩽̹ f ( S ( N )) .