More ZFC inequalities between cardinal invariants Vera Fischer - PowerPoint PPT Presentation

More ZFC inequalities between cardinal invariants Vera Fischer University of Vienna January 2019 Vera Fischer (University of Vienna) More ZFC inequalities January 2019 1 / 46

Content Outline: Higher Analogues Eventual difference and a e ( κ ) , a p ( κ ) , a g ( κ ) ; 1 Generalized Unsplitting and Domination; 2 Vera Fischer (University of Vienna) More ZFC inequalities January 2019 2 / 46

Eventual Difference Eventual Difference Almost disjointness a ( κ ) is the min size of a max almost disjoint A ⊆ [ κ ] κ of size ≥ κ Relatives a e ( κ ) is the min size of max, eventually different family F ⊆ κ κ , a p ( κ ) is the min size of a max, eventually different family F ⊆ S ( κ ) := { f ∈ κ κ : f is bijective } , a g ( κ ) is the min size of a max, almost disjoint subgroup of S ( κ ) . Vera Fischer (University of Vienna) More ZFC inequalities January 2019 3 / 46

Eventual Difference What we still do not know... Even though Con ( a < a g ) , both the consistency of a g < a , as well as the inequality a ≤ a g (in ZFC) are open. Vera Fischer (University of Vienna) More ZFC inequalities January 2019 4 / 46

Eventual Difference Roitman Problem Is it consistent that d < a ? Yes, if ℵ 1 < d (Shelah’s template construction). Open, if ℵ 1 = d . Is it consistent that d = ℵ 1 < a g ? Vera Fischer (University of Vienna) More ZFC inequalities January 2019 5 / 46

Eventual Difference One of the major differences between a and its relatives, is their relation to non ( M ) . While a and non ( M ) are independent, non ( M ) ≤ a e , a p , a g (Brendle, Spinas, Zhang), Thus in particular, consistently d = ℵ 1 < a g = ℵ 2 . Vera Fischer (University of Vienna) More ZFC inequalities January 2019 6 / 46

Eventual Difference Uniformity of the Meager Ideal: Higher Analogue For κ regular uncountable, define nm ( κ ) to be the least size of a family F ⊆ κ κ such that ∀ g ∈ κ κ ∃ f ∈ F with |{ α ∈ κ : f ( α ) = g ( α ) }| = κ . Theorem (Hyttinen) If κ is a successor, then mn ( κ ) = b ( κ ) . Vera Fischer (University of Vienna) More ZFC inequalities January 2019 7 / 46

Eventual Difference Theorem (Blass, Hyttinen, Zhang) Let κ be regular uncountable. Then b ( κ ) ≤ a ( κ ) , a e ( κ ) , a p ( κ ) , a g ( κ ) ; Corollary Thus for κ successors, nm ( κ ) = b ( κ ) ≤ a ( κ ) , a e ( κ ) , a p ( κ ) , a g ( κ ) . Vera Fischer (University of Vienna) More ZFC inequalities January 2019 8 / 46

Eventual Difference Roitman in the Uncountable Theorem (Blass, Hyttinen and Zhang) Let κ ≥ ℵ 1 be regular uncountable. Then d ( κ ) = κ + ⇒ a ( κ ) = κ + . Vera Fischer (University of Vienna) More ZFC inequalities January 2019 9 / 46

Eventual Difference Roitman in the Uncountable The cofinitary groups analogue Clearly, the result does not have a cofinitary group analogue for κ = ℵ 0 , since d = ℵ 1 < a g = a g ( ℵ 0 ) = ℵ 2 is consistent. Nevertheless the question remains of interest for uncountable κ : Is it consistent that d ( κ ) = κ + ⇒ a g ( κ ) = κ + ? Vera Fischer (University of Vienna) More ZFC inequalities January 2019 10 / 46

Eventual Difference Club unboundedness Theorem (Raghavan, Shelah, 2018) Let κ be regular uncountable. Then b ( κ ) = κ + ⇒ a ( κ ) = κ + . Vera Fischer (University of Vienna) More ZFC inequalities January 2019 11 / 46

Eventual Difference Club unboundedness Let κ be regular uncountable. For f , g ∈ κ κ we say that f ≤ cl g iff 1 { α < κ : g ( α ) < f ( α ) } is non-stationary. F ⊆ κ κ is ≤ cl -unbounded if ¬∃ g ∈ κ κ ∀ f ∈ F ( f ≤ cl g ) . 2 b cl ( κ ) = min {| F | : F ⊆ κ κ and F is cl-unbounded } 3 Vera Fischer (University of Vienna) More ZFC inequalities January 2019 12 / 46

Eventual Difference Theorem (Cummings, Shelah) If κ is regular uncountable then b ( κ ) = b cl ( κ ) . Vera Fischer (University of Vienna) More ZFC inequalities January 2019 13 / 46

Eventual Difference Higher eventually different analogues Theorem(F ., D. Soukup, 2018) Suppose κ = λ + for some infinite λ and b ( κ ) = κ + . Then a e ( κ ) = a p ( κ ) = κ + . If in addition 2 < λ = λ , then a g ( κ ) = κ + . Remark The case of a e ( κ ) has been considered earlier. The above is a strengthening of each of the following: d ( κ ) = κ + ⇒ a e ( κ ) = κ + for κ successor (Blass, Hyttinen, Zhang) b ( κ ) = κ + ⇒ a e ( κ ) = κ + proved by Hyttinen under additional assumptions. Vera Fischer (University of Vienna) More ZFC inequalities January 2019 14 / 46

Eventual Difference Outline: b ( κ ) = κ + ⇒ a e ( κ ) = κ + For each λ : λ ≤ α < λ + = κ fix a bijection d α : α → λ . For each δ : λ + = κ ≤ δ < κ + fix a bijection e δ : κ → δ . Let { f δ : δ < κ + } witness b cl ( κ ) = κ + . Vera Fischer (University of Vienna) More ZFC inequalities January 2019 15 / 46

Eventual Difference We will define functions h δ , ζ ∈ κ κ for δ < κ + , ζ < λ by induction on δ , simultaneously for all ζ < λ . Thus, suppose we have defined h δ ′ , ζ for δ ′ < δ , ζ < λ . Let µ < κ . We want to define h δ , ζ ( µ ) for each ζ ∈ λ . Let H δ ( µ ) = { h δ ′ , ζ ′ : δ ′ ∈ ran ( e δ ↾ µ ) , ζ ′ ∈ λ } . Then, since e δ : κ → δ is a bijection, | ran ( e δ ↾ µ ) | ≤ λ and so H δ ( µ ) , as well as H δ ( µ ) = { h ( µ ) : h ∈ H δ ( µ ) } are of size < κ . Vera Fischer (University of Vienna) More ZFC inequalities January 2019 16 / 46

Eventual Difference Then define: f ∗ δ ( µ ) = max { f δ ( µ ) , min { α ∈ κ : | α \ H δ ( µ ) | = λ }} < κ . Now, | f ∗ δ ( µ ) \ H δ ( µ ) | = λ and so, we have enough space to define the values h δ , ζ ( µ ) distinct for all ζ < λ . More precisely, for each ζ < λ , define h δ , ζ ( µ ) := β where β is such that δ ( µ ) [ β ∩ ( f ∗ d f ∗ δ ( µ ) \ H δ ( µ ))] is of order type ζ . We say that h δ , ζ ( µ ) is the ζ -th element of f ∗ δ ( µ ) \ H δ ( µ ) with respect to d f ∗ δ ( µ ) . Vera Fischer (University of Vienna) More ZFC inequalities January 2019 17 / 46

Eventual Difference Claim: { h δ , ζ } δ < κ + , ζ < λ is κ -e.d. Case 1: Fix δ < κ + . If ζ � = ζ ′ , then by definition h δ , ζ ( µ ) � = h δ , ζ ′ ( µ ) for each µ < κ . Case 2: Let δ ′ < δ < κ + and ζ , ζ ′ < λ be given. Since e δ : κ → δ is a bijection, there is µ 0 ∈ κ such that δ ′ ∈ range ( e δ ↾ µ 0 ) . But then for each µ ≥ µ 0 , h δ ′ , ζ ′ ∈ H δ ( µ ) and so h δ ′ , ζ ′ ( µ ) ∈ H δ ( µ ) . However h δ , ζ ∈ κ \ H δ ( µ ) and so h δ ′ , ζ ′ ( µ ) � = h δ , ζ ( µ ) . Vera Fischer (University of Vienna) More ZFC inequalities January 2019 18 / 46

Eventual Difference Claim: { h δ , ζ } δ < κ + , ζ < λ is κ -med. Let h ∈ κ κ and δ < κ + such that S = { µ ∈ κ : h ( µ ) < f δ ( µ ) } is stationary. There is stationary S 0 ⊆ S such that h ( µ ) ∈ H δ ( µ ) for all µ ∈ S 0 , or 1 h ( µ ) / ∈ H δ ( µ ) for all µ ∈ S 0 . 2 We will see that in either case, there are δ , ζ such that h δ , ζ ( µ ) = h ( µ ) for stationarily many µ ∈ S 0 . Vera Fischer (University of Vienna) More ZFC inequalities January 2019 19 / 46

Eventual Difference Case 1: h ( µ ) ∈ H δ ( µ ) for all µ ∈ S 0 Recall: H δ ( µ ) = { h δ ′ , ζ ′ : δ ′ ∈ ran ( e δ ↾ µ ) , ζ ′ ∈ λ } , and H δ ( µ ) = { h ( µ ) : h ∈ H δ ( µ ) } . Now: For each µ ∈ S 0 there are η µ < µ , ζ µ < λ such that h ( µ ) = h e δ ( η µ ) , ζ µ ( µ ) . By Fodor’s Lemma we can find a stationary S 1 ⊆ S 0 such that for all µ ∈ S 1 , η µ = η for some η < µ . Then for δ ′ = e δ ( η ) we can find stationarily many µ ∈ S 1 such that ζ µ = ζ ′ for some ζ ′ , and so for stationarily many µ in S 1 we have h ( µ ) = h δ ′ , ζ ′ ( µ ) . Vera Fischer (University of Vienna) More ZFC inequalities January 2019 20 / 46

Eventual Difference Case 2: h ( µ ) / ∈ H δ ( µ ) for all µ ∈ S 0 Recall: f ∗ δ ( µ ) = max { f δ ( µ ) , min { α ∈ κ : | α \ H δ ( µ ) | = λ }} < κ Now: For each µ ∈ S 0 , since h ( µ ) < f δ ( µ ) and f δ ( µ ) ≤ f ∗ δ ( µ ) , we have h ( µ ) ∈ f ∗ δ ( µ ) \ H δ ( µ ) . Thus, for each µ ∈ S 0 \ ( λ + 1 ) there is ζ µ < λ ≤ µ such that h ( µ ) is the ζ µ -th element of f ∗ δ ( µ ) \ H δ ( µ ) with respect to d f ∗ δ ( µ ) . By Fodor’s Lemma, there is a stationary S 1 ⊆ S 0 such that for each µ ∈ S 1 , ζ = ζ µ for some ζ and so for all µ ∈ S 1 we have h ( µ ) = h δ , ζ ( µ ) . Vera Fischer (University of Vienna) More ZFC inequalities January 2019 21 / 46

Eventual Difference Question Is it true that b ( κ ) = κ + implies that a e ( κ ) = a p ( κ ) = κ + if κ is not a successor? Vera Fischer (University of Vienna) More ZFC inequalities January 2019 22 / 46