Combinatorics, Cardinal Characteristics of the Continuum, and the Colouring Calculus

Transcription

1 Combinatorics, Cardinal Characteristics of the Continuum, and the Colouring Calculus 03E05, 03E17 & 03E02 Thilo Weinert Ben-Gurion-University of the Negev Joint work with William Chen and Chris Lambie-Hanson 10th Young Set Theory Workshop, Edinburgh, Thursday, 13 th July 2017, 11:30-12:30

2 1 Combinatorial Principles 2 Cardinal Characteristics 3 Hybridisation 4 A Diagram 5 The Colouring Calculus 6 Polarised Partition Relations 7 New Results 8 A Table 9 Questions

3 Combinatorial Principles Definition (Cantor, 1878) CH Problem (Souslin, 1920) Is every dense complete linear order without endpoints, for which every collection of disjoint open intervals is countable, order-isomorphic to the real line? Definition Souslin s Hypothesis(SH): Yes.

4 Combinatorial Principles Definition (Jensen, 1972) says that there is a sequence S α α < ω 1 with S α α for all α < ω 1 and S ω 1 ({α < ω 1 S α = S α} is stationary.) Theorem (Jensen, 1972) L =. Corollary ZFC + is consistent. Theorem (Jensen, 1972) ( SH).

5 Combinatorial Principles Theorem (Jensen, 1974) ZFC + CH + SH is consistent. Corollary ZFC + CH + is consistent. Definition (Ostaszewski, 1976) says that there is a sequence S α α < ω 1 with S α α for all α < ω 1 and S ω 1 α < ω 1 (S α S). Definition (Broverman, Ginsburg, Kunen & Tall, 1978) says that there is a family F [[ω1 ] ω ] ω 1 such that S ω 1 X F (X S)

6 Combinatorial Principles Exercise = =, and = CH =. Theorem (Devlin, 1979) CH Corollary ZFC + CH + is consistent. Theorem (Shelah, 1980) ZFC + + CH is consistent.

7 Combinatorial Principles SH CH

8 Cardinal Characteristics For a cardinal κ and functions f, g κ κ, we write f g to express that {α < κ f (α) < g(α)} is bounded. Definition b κ := min { F F κ κ g κ κ f F (g f )}. b := b ω. b is called the unbounding number. Definition d κ := min { F F κ κ g κ κ f F (f g)}. d := d ω. d is called the dominating number.

9 Cardinal Characteristics For a cardinal κ and x, y κ, we say that x splits y if both y x and y \ x have cardinality κ. Definition The splitting number s κ is the minimal size of a subfamily F of (κ) having the property that for every y (κ) there is an x F splitting y. s := s ω. Definition The reaping(refining(unsplitting)) number r κ is the minimal size of a subfamily F of (κ) for which there is no single x (κ) splitting all y F. r := r ω.

10 Cardinal Characteristics A tower is a sequence F α α < γ of infinite sets of natural numbers such that for α < β < γ the set F β \ F α is finite. It is extendible if there is an infinite set R of natural numbers such that for all α < γ the set R \ F α is finite. Definition t is the minimal length of an unextendible tower. Definition For an ideal I on a set X, the covering number, cov(i ) denotes the minimal number of sets in I needed to cover all of X. the additivity number, add(i ) denotes the minimal number of sets in I whose union lies outside of I.

11 Hybridisation Theorem (Truss, 1983) = min(cov(m), cov(n )) = ℵ1. Theorem (Miyamoto, 1994, unpublished) SH = cov(m) = ℵ1. Definition (Fuchino, Shelah & Soukup, 1997) is the minimal size of a family F [ω1 ] ω such that for every S [ω 1 ] ω 1 there is an X F with X S.

12 Hybridisation Theorem (Juhasz unpublished; Fuchino, Shelah & Soukup, 1997) ZFC + + cov(m) = ℵ 2 is consistent. Theorem (Brendle, Malliaris & Shelah, 2006 & 2013) t. Theorem (Brendle, 2006) add(n ). Theorem (Brendle, 2006) ZFC + + cov(n ) = ℵ 2 is consistent.

13 A Diagram c i cof(n ) r Q r d cov(n ) b cov(m) s add(m) s Q add(n ) t ℵ 1

14 The Colouring Calculus Notation (Erdős & Rado, 1956) α (β 0,..., β k ) n says that for every colouring χ of the n-tuples of a set X of size α with ran(χ) = k + 1, there is an i k and a subset H X of size β i such that χ [ [H] n] = {i}. Notation α (β) n k abbreviates α (β,..., β) }{{} n. k times

15 The Colouring Calculus Theorem (Sierpinski, 1933) ω 1 (ω 1 ) 2 2. Theorem (Erdős & Rado, 1956) r(ωm, n) = ωr(i m, L n ) for all natural numbers m and n. Theorem (Hajnal, 1960) CH = ω 1 (ω 1, ω + 2) 2. Theorem (Hajnal, 1971) GCH = (κ + ) 2 ((κ + ) 2, 3) 2 for regular κ.

16 The Colouring Calculus Theorem (Erdős & Hajnal, 1971) GCH = κ + κ (κ + κ, 3) 2 for cardinals κ. Theorem (Erdős & Hajnal, 1971) (κ + ) 2 (α, 3) 2 for all cardinals κ and all α < (κ + ) 2. Theorem (Baumgartner, 1972) r(κm, n) = κr(i m, L n ) for all cardinals κ and all m, n < ω. Theorem (Baumgartner & Hajnal, 1973) n < ω α < ω 1 (ω 1 (α) 2 n).

17 The Colouring Calculus Theorem (Baumgartner, 1975) ω1 2 (ω2 1, 3)2 = SH. Theorem (Todorcevic, 1983) α < ω 1 : ω 1 (ω 1, α) 2 is consistent. Theorem (Erdős, Hajnal, Mate & Rado, 1984) If κ is regular and uncountable, then κ (κ, ω + 1) 2. Theorem (Takahashi, ) = ℵ1 = ω1 2 (ω2 1, 3)2.

18 The Colouring Calculus Theorem (Takahashi, ) = ℵ1 = d = ω 1 ω (ω 1 ω, 3) 2. Theorem (Baumgartner & Hajnal, ) If κ is regular and 2 κ = κ +, then (κ + ) 2 (κ + κ, 4) 2. Theorem (Baumgartner & Hajnal, ) ω1 2 (ω 1ω, 3, 3) 2. Theorem (Baumgartner, 1989) MA ℵ1 = n < ω : ω 1 ω (ω 1 ω, n) 2

19 The Colouring Calculus Theorem (Todorcevic, 1989) b = ℵ 1 = ω 1 (ω 1, ω + 2) 2. Theorem (J. Larson, 1998) If κ is regular and d(κ) = κ +, then κ + κ (κ + κ, 3) 2. Theorem (J. Larson, 1998) If κ is regular and d(κ) = κ +, then (κ + ) 2 ((κ + ) 2, 3) 2. Theorem (Raghavan & Todorcevic, 2016) α < ω 1 (b (b, α) 2 ) is consistent, relative to the existence of a measurable cardinal.

20 Polarised Partition Relations Notation ( ) α β ( ) γ δ says that for all sets A of size α, B of size β and every colouring χ : A B 2 there are C A of size γ and D B of size δ such that χ [ C D ] = 1.

21 Polarised Partition Relations Theorem (Garti & Shelah, 2014) If κ is regular and κ < λ < s κ, then ( ) ( ) λ λ cf(λ) κ. κ κ Theorem (Garti & Shelah, 2014) If κ is regular and r κ < cf(λ) < 2 κ, then ( ) ( ) λ λ. κ κ

22 New Results Theorem (Lambie-Hanson & W., 2016) b κ = κ + = κ = κ+ κ (κ + κ, 3) 2 for regular κ. Theorem (Chen & W., 2016) κ = κ+ = κ + (κ +, ω + 2) 2.

23 New Results Statement (J. Larson, 1998) It would be interesting to know if the hypothesis ( for (κ + ) 2 (κ + κ, 4) 2) can be weakened to the existence of a short scale in κ+ κ +.

24 New Results Statement (J. Larson, 1998) It would be interesting to know if the hypothesis ( for (κ + ) 2 (κ + κ, 4) 2) can be weakened to the existence of a short scale in κ+ κ +. Theorem (Chen & W., 2017) d κ = κ + = (κ + ) 2 (κ + κ, 4) 2 for regular κ.

25 New Results Statement (J. Larson, 1998) It would be interesting to know if the hypothesis ( for (κ + ) 2 (κ + κ, 4) 2) can be weakened to the existence of a short scale in κ+ κ +. Theorem (Chen & W., 2017) d κ = κ + = (κ + ) 2 (κ + κ, 4) 2 for regular κ. Theorem (Chen & W., 2017) b κ = κ + = κ = (κ+ ) 2 (κ + κ, 4) 2 for regular κ.

26 A Table CH d = ℵ 1 = ω 1 (ω 1, ω + 2) 2 Hajnal 1960 ω1 2 (ω2 1, 3)2 ω 1 ω (ω 1 ω, 3) 2 ω1 2 (ω 1ω, 4) 2 d = ℵ 1 b = ℵ 1 = = ℵ1 b = ℵ 1 ZFC

27 A Table CH d = ℵ 1 = ω 1 (ω 1, ω + 2) 2 ω1 2 (ω2 1, 3)2 ω 1 ω (ω 1 ω, 3) 2 Hajnal Hajnal Erdős ω1 2 (ω 1ω, 4) 2 Hajnal d = ℵ 1 b = ℵ 1 = = ℵ1 b = ℵ 1 ZFC

28 A Table CH d = ℵ 1 = ω 1 (ω 1, ω + 2) 2 ω1 2 (ω2 1, 3)2 ω 1 ω (ω 1 ω, 3) 2 Hajnal Hajnal Erdős ω1 2 (ω 1ω, 4) 2 Hajnal d = ℵ 1 b = ℵ 1 = = ℵ1 b = ℵ 1 ZFC Todorcevic 1983

29 A Table ω 1 (ω 1, ω + 2) 2 ω1 2 (ω2 1, 3)2 ω 1 ω (ω 1 ω, 3) 2 ω1 2 (ω 1ω, 4) 2 Hajnal Hajnal Erdős Baumgartner CH Hajnal Hajnal d = ℵ 1 = d = ℵ 1 b = ℵ 1 = = ℵ1 b = ℵ 1 ZFC Todorcevic 1983

30 A Table ω 1 (ω 1, ω + 2) 2 ω1 2 (ω2 1, 3)2 ω 1 ω (ω 1 ω, 3) 2 ω1 2 (ω 1ω, 4) 2 Hajnal Hajnal Erdős Baumgartner CH Hajnal Hajnal d = ℵ 1 = Takahashi d = ℵ 1 b = ℵ 1 = = ℵ1 b = ℵ 1 ZFC Todorcevic 1983

31 A Table ω 1 (ω 1, ω + 2) 2 ω1 2 (ω2 1, 3)2 ω 1 ω (ω 1 ω, 3) 2 ω1 2 (ω 1ω, 4) 2 Hajnal Hajnal Erdős Baumgartner CH Hajnal Hajnal d = ℵ 1 = O Takahashi d = ℵ 1 b = ℵ 1 = O Takahashi = ℵ1 b = ℵ 1 ZFC Todorcevic 1983

32 A Table ω 1 (ω 1, ω + 2) 2 ω1 2 (ω2 1, 3)2 ω 1 ω (ω 1 ω, 3) 2 ω1 2 (ω 1ω, 4) 2 Hajnal Hajnal Erdős Baumgartner CH Hajnal Hajnal d = ℵ 1 = O O Takahashi d = ℵ 1 O b = ℵ 1 = O O Takahashi = ℵ1 b = ℵ 1 ZFC Todorcevic Todorcevic 1983

33 A Table ω 1 (ω 1, ω + 2) 2 ω1 2 (ω2 1, 3)2 ω 1 ω (ω 1 ω, 3) 2 ω1 2 (ω 1ω, 4) 2 Hajnal Hajnal Erdős Baumgartner CH Hajnal Hajnal d = ℵ 1 = O O Takahashi d = ℵ 1 O b = ℵ 1 = O O Takahashi = ℵ1 b = ℵ 1 Todorcevic ZFC Todorcevic Baumgartner Baumgartner

34 A Table ω 1 (ω 1, ω + 2) 2 ω1 2 (ω2 1, 3)2 ω 1 ω (ω 1 ω, 3) 2 ω1 2 (ω 1ω, 4) 2 Hajnal Hajnal Erdős Baumgartner CH Hajnal Hajnal d = ℵ 1 = O O Takahashi Larson Larson d = ℵ 1 O b = ℵ 1 = O O Takahashi = ℵ1 b = ℵ 1 Todorcevic ZFC Todorcevic Baumgartner Baumgartner

35 A Table ω 1 (ω 1, ω + 2) 2 ω1 2 (ω2 1, 3)2 ω 1 ω (ω 1 ω, 3) 2 ω1 2 (ω 1ω, 4) 2 Hajnal Hajnal Erdős Baumgartner CH Hajnal Hajnal d = ℵ 1 = O O Takahashi Larson Larson d = ℵ 1 O Lambie-Hanson b = ℵ 1 = O O W Takahashi = ℵ1 b = ℵ 1 Todorcevic ZFC Todorcevic Baumgartner Baumgartner

36 A Table ω 1 (ω 1, ω + 2) 2 ω1 2 (ω2 1, 3)2 ω 1 ω (ω 1 ω, 3) 2 ω1 2 (ω 1ω, 4) 2 Hajnal Hajnal Erdős Baumgartner CH Hajnal Hajnal d = ℵ 1 = O O Takahashi Larson Larson d = ℵ 1 O Lambie-Hanson b = ℵ 1 = O O W Chen Takahashi = ℵ1 W b = ℵ 1 Todorcevic ZFC Todorcevic Baumgartner Baumgartner

37 A Table ω 1 (ω 1, ω + 2) 2 ω1 2 (ω2 1, 3)2 ω 1 ω (ω 1 ω, 3) 2 ω1 2 (ω 1ω, 4) 2 Hajnal Hajnal Erdős Baumgartner CH Hajnal Hajnal d = ℵ 1 = O O Takahashi O Larson Larson Chen d = ℵ 1 O W Lambie-Hanson b = ℵ 1 = O O W Chen Takahashi = ℵ1 W b = ℵ 1 Todorcevic ZFC Todorcevic Baumgartner Baumgartner

38 A Table ω 1 (ω 1, ω + 2) 2 ω1 2 (ω2 1, 3)2 ω 1 ω (ω 1 ω, 3) 2 ω1 2 (ω 1ω, 4) 2 Hajnal Hajnal Erdős Baumgartner CH Hajnal Hajnal d = ℵ 1 = O O Takahashi O Larson Larson Chen d = ℵ 1 O W Lambie-Hanson Chen b = ℵ 1 = O O W. W Chen Takahashi = ℵ1 W b = ℵ 1 Todorcevic ZFC Todorcevic Baumgartner Baumgartner

39 A Table ω 1 (ω 1, ω + 2) 2 ω1 2 (ω2 1, 3)2 ω 1 ω (ω 1 ω, 3) 2 ω1 2 (ω 1ω, 4) 2 Hajnal Hajnal Erdős Baumgartner CH Hajnal Hajnal d = ℵ 1 = O O Takahashi O Larson Larson Chen d = ℵ 1 O W Lambie-Hanson Chen b = ℵ 1 = O O W. W Chen Takahashi = ℵ1 W.?? 2016 b = ℵ 1 Todorcevic??? ZFC Todorcevic? Baumgartner Baumgartner

40 Questions Question Is ZFC + α < ω 2 1 (ω2 1 (α, 3, 3)2 ) consistent? Question (Baumgartner & Hajnal, ) Is ZFC +ω1 2 (ω 1ω, 3, 3, 3) 2 consistent?

41 Questions Question (Jean Larson) Is ZFC +ω1 2 (ω2 1, 3)2 consistent? Question Is ZFC +b = ℵ 1 + ω1 2 (ω2 1, 3)2 consistent? Question Is ZFC +b = ℵ 1 + ω 1 ω (ω 1 ω, 3) 2 consistent? Question Is ZFC + = ℵ 1 + ω 1 ω (ω 1 ω, 3) 2 consistent?

42 Questions Question Is ZFC +b = ℵ 1 + ω 2 1 (ω 1ω, 4) 2 consistent? Question Is ZFC + = ℵ 1 + ω 2 1 (ω 1ω, 4) 2 consistent? Question Is ZFC + κ(κ is regular, b κ = κ + and κ + (κ +, κ + 2) 2 ) consistent?

43 Questions Question Is ZFC + MA ℵ1 + α < ω 1 : ω 1 (ω 1, α) 2 consistent? Question (Brendle, 2006) Is ZFC + + s > ℵ 1 consistent? Question Is ZFC + + s Q > ℵ 1 consistent? Question (Juhász,?) Is ZFC + + SH consistent?

44 Questions Thank you for listening!