Review: definable graphs and definable chromatic numbers The G 0 graph and the G 0 -dichotomy What is the Borel chromatic number of G 0 ? Separating Borel chromatic number of closed graphs Definition Let ( s n ) n ∈ ω be a dense sequence of elements of 2 <ω such that | s n | = n for all n ∈ ω , and every t ∈ 2 <ω has an extension of the form s n . The G 0 -graph is the graph on 2 ω defined by G 0 . = { ( s � n 0 � x , s � n 1 � x ) | n ∈ ω ∧ x ∈ 2 ω } . 20 / 132
Review: definable graphs and definable chromatic numbers The G 0 graph and the G 0 -dichotomy What is the Borel chromatic number of G 0 ? Separating Borel chromatic number of closed graphs Definition Let ( s n ) n ∈ ω be a dense sequence of elements of 2 <ω such that | s n | = n for all n ∈ ω , and every t ∈ 2 <ω has an extension of the form s n . The G 0 -graph is the graph on 2 ω defined by G 0 . = { ( s � n 0 � x , s � n 1 � x ) | n ∈ ω ∧ x ∈ 2 ω } . Lemma The chromatic number of G 0 is 2 and the Borel chromatic number of G 0 is uncountable. 21 / 132
Review: definable graphs and definable chromatic numbers The G 0 graph and the G 0 -dichotomy What is the Borel chromatic number of G 0 ? Separating Borel chromatic number of closed graphs Definition Let ( s n ) n ∈ ω be a dense sequence of elements of 2 <ω such that | s n | = n for all n ∈ ω , and every t ∈ 2 <ω has an extension of the form s n . The G 0 -graph is the graph on 2 ω defined by G 0 . = { ( s � n 0 � x , s � n 1 � x ) | n ∈ ω ∧ x ∈ 2 ω } . Lemma The chromatic number of G 0 is 2 and the Borel chromatic number of G 0 is uncountable. In fact, χ B ( G 0 ) ≥ cov ( M ) . 22 / 132
Review: definable graphs and definable chromatic numbers The G 0 graph and the G 0 -dichotomy What is the Borel chromatic number of G 0 ? Separating Borel chromatic number of closed graphs Definition Let ( s n ) n ∈ ω be a dense sequence of elements of 2 <ω such that | s n | = n for all n ∈ ω , and every t ∈ 2 <ω has an extension of the form s n . The G 0 -graph is the graph on 2 ω defined by G 0 . = { ( s � n 0 � x , s � n 1 � x ) | n ∈ ω ∧ x ∈ 2 ω } . Lemma The chromatic number of G 0 is 2 and the Borel chromatic number of G 0 is uncountable. In fact, χ B ( G 0 ) ≥ cov ( M ) . For proof see (see [KST99]. 23 / 132
Review: definable graphs and definable chromatic numbers The G 0 graph and the G 0 -dichotomy What is the Borel chromatic number of G 0 ? Separating Borel chromatic number of closed graphs Theorem (Kechris-Solecki-Todorcevic G 0 -dichotomy, [KST99]) Let X be a Polish space and G be an analytic graph on X . Then exactly one of the following holds: 24 / 132
Review: definable graphs and definable chromatic numbers The G 0 graph and the G 0 -dichotomy What is the Borel chromatic number of G 0 ? Separating Borel chromatic number of closed graphs Theorem (Kechris-Solecki-Todorcevic G 0 -dichotomy, [KST99]) Let X be a Polish space and G be an analytic graph on X . Then exactly one of the following holds: (a) either χ B ( G ) ≤ ℵ 0 , 25 / 132
Review: definable graphs and definable chromatic numbers The G 0 graph and the G 0 -dichotomy What is the Borel chromatic number of G 0 ? Separating Borel chromatic number of closed graphs Theorem (Kechris-Solecki-Todorcevic G 0 -dichotomy, [KST99]) Let X be a Polish space and G be an analytic graph on X . Then exactly one of the following holds: (a) either χ B ( G ) ≤ ℵ 0 , or (b) there is a continuous homomorphism from G 0 to G . In this case χ B ( G 0 ) ≤ χ B ( G ) . 26 / 132
Review: definable graphs and definable chromatic numbers The G 0 graph and the G 0 -dichotomy What is the Borel chromatic number of G 0 ? Separating Borel chromatic number of closed graphs Theorem (Kechris-Solecki-Todorcevic G 0 -dichotomy, [KST99]) Let X be a Polish space and G be an analytic graph on X . Then exactly one of the following holds: (a) either χ B ( G ) ≤ ℵ 0 , or (b) there is a continuous homomorphism from G 0 to G . In this case χ B ( G 0 ) ≤ χ B ( G ) . The G 0 -dichotomy implies many known dichotomies in descritive set theory such as the perfect set property or the Silver’s dichotomy on co-analytic equivalence relations (see [Mil12]). 27 / 132
Review: definable graphs and definable chromatic numbers The G 0 graph and the G 0 -dichotomy What is the Borel chromatic number of G 0 ? Separating Borel chromatic number of closed graphs We want to understand the relationship between χ B ( G 0 ) and the various cardinal invariants in the Cichón’s diagram. 28 / 132
Review: definable graphs and definable chromatic numbers The G 0 graph and the G 0 -dichotomy What is the Borel chromatic number of G 0 ? Separating Borel chromatic number of closed graphs We want to understand the relationship between χ B ( G 0 ) and the various cardinal invariants in the Cichón’s diagram. First, notice that G has a perfect clique ⇒ χ B ( G ) = 2 ℵ 0 . 29 / 132
Review: definable graphs and definable chromatic numbers The G 0 graph and the G 0 -dichotomy What is the Borel chromatic number of G 0 ? Separating Borel chromatic number of closed graphs We want to understand the relationship between χ B ( G 0 ) and the various cardinal invariants in the Cichón’s diagram. First, notice that G has a perfect clique ⇒ χ B ( G ) = 2 ℵ 0 . Now, G 0 belongs to the class of closed graphs on the Cantor space. 30 / 132
Review: definable graphs and definable chromatic numbers The G 0 graph and the G 0 -dichotomy What is the Borel chromatic number of G 0 ? Separating Borel chromatic number of closed graphs We want to understand the relationship between χ B ( G 0 ) and the various cardinal invariants in the Cichón’s diagram. First, notice that G has a perfect clique ⇒ χ B ( G ) = 2 ℵ 0 . Now, G 0 belongs to the class of closed graphs on the Cantor space. We address the question of what are possible Borel chromatic numbers of closed graphs without perfect cliques in Polish spaces. 31 / 132
Review: definable graphs and definable chromatic numbers The G 0 graph and the G 0 -dichotomy What is the Borel chromatic number of G 0 ? Separating Borel chromatic number of closed graphs We want to understand the relationship between χ B ( G 0 ) and the various cardinal invariants in the Cichón’s diagram. First, notice that G has a perfect clique ⇒ χ B ( G ) = 2 ℵ 0 . Now, G 0 belongs to the class of closed graphs on the Cantor space. We address the question of what are possible Borel chromatic numbers of closed graphs without perfect cliques in Polish spaces. We also would like to compare them to other cardinal characteristics of the continuum. 32 / 132
Review: definable graphs and definable chromatic numbers The G 0 graph and the G 0 -dichotomy What is the Borel chromatic number of G 0 ? Separating Borel chromatic number of closed graphs First, in the model obtained by adding κ Cohen reals with finite support interation 33 / 132
Review: definable graphs and definable chromatic numbers The G 0 graph and the G 0 -dichotomy What is the Borel chromatic number of G 0 ? Separating Borel chromatic number of closed graphs First, in the model obtained by adding κ Cohen reals with finite support interation cov ( M ) = χ B ( G ) = κ = 2 ℵ 0 , for any analytic graph G on a Polish space with uncountable Borel chromatic number 34 / 132
Review: definable graphs and definable chromatic numbers The G 0 graph and the G 0 -dichotomy What is the Borel chromatic number of G 0 ? Separating Borel chromatic number of closed graphs First, in the model obtained by adding κ Cohen reals with finite support interation cov ( M ) = χ B ( G ) = κ = 2 ℵ 0 , for any analytic graph G on a Polish space with uncountable Borel chromatic number (by the the well-known fact that Cohen forcing increases cov ( M ) and cov ( M ) ≤ χ B ( G 0 ) ≤ χ B ( G ) ≤ 2 ℵ 0 for G as above). 35 / 132
Review: definable graphs and definable chromatic numbers The G 0 graph and the G 0 -dichotomy What is the Borel chromatic number of G 0 ? Separating Borel chromatic number of closed graphs First, in the model obtained by adding κ Cohen reals with finite support interation cov ( M ) = χ B ( G ) = κ = 2 ℵ 0 , for any analytic graph G on a Polish space with uncountable Borel chromatic number (by the the well-known fact that Cohen forcing increases cov ( M ) and cov ( M ) ≤ χ B ( G 0 ) ≤ χ B ( G ) ≤ 2 ℵ 0 for G as above). This is a situation where the Borel chromatic number of all analytic graphs is either countable or as big as the continuum. We will first see that this is independent of ZFC. 36 / 132
Review: definable graphs and definable chromatic numbers The G 0 graph and the G 0 -dichotomy What is the Borel chromatic number of G 0 ? Separating Borel chromatic number of closed graphs Theorem Let X be a Polish space and G = ( X , E ) be a closed graph and suppose that CH holds in the ground model. 37 / 132
Review: definable graphs and definable chromatic numbers The G 0 graph and the G 0 -dichotomy What is the Borel chromatic number of G 0 ? Separating Borel chromatic number of closed graphs Theorem Let X be a Polish space and G = ( X , E ) be a closed graph and suppose that CH holds in the ground model. Then either G has a perfect clique 38 / 132
Review: definable graphs and definable chromatic numbers The G 0 graph and the G 0 -dichotomy What is the Borel chromatic number of G 0 ? Separating Borel chromatic number of closed graphs Theorem Let X be a Polish space and G = ( X , E ) be a closed graph and suppose that CH holds in the ground model. Then either G has a perfect clique or χ B ( G ) ≤ ℵ 1 in the Sacks model. 39 / 132
Review: definable graphs and definable chromatic numbers The G 0 graph and the G 0 -dichotomy What is the Borel chromatic number of G 0 ? Separating Borel chromatic number of closed graphs Theorem Let X be a Polish space and G = ( X , E ) be a closed graph and suppose that CH holds in the ground model. Then either G has a perfect clique or χ B ( G ) ≤ ℵ 1 in the Sacks model. In this model, the continuum is 2 ℵ 0 = ℵ 2 . It remains open if the same holds for the classe of analytic graphs on Polish spaces. 40 / 132
Review: definable graphs and definable chromatic numbers The G 0 graph and the G 0 -dichotomy What is the Borel chromatic number of G 0 ? Separating Borel chromatic number of closed graphs In the Cohen model: if the Borel chromatic number of an analytic graph is uncountable, then it has the cardinality of the continuum — they are all equally big. 41 / 132
Review: definable graphs and definable chromatic numbers The G 0 graph and the G 0 -dichotomy What is the Borel chromatic number of G 0 ? Separating Borel chromatic number of closed graphs In the Cohen model: if the Borel chromatic number of an analytic graph is uncountable, then it has the cardinality of the continuum — they are all equally big. In the Sacks model: if a closed graph does not have perfect clique, then its Borel chromatic number is at most ℵ 1 in the Sacks model — they are all equally small. 42 / 132
Review: definable graphs and definable chromatic numbers The G 0 graph and the G 0 -dichotomy What is the Borel chromatic number of G 0 ? Separating Borel chromatic number of closed graphs In the Cohen model: if the Borel chromatic number of an analytic graph is uncountable, then it has the cardinality of the continuum — they are all equally big. In the Sacks model: if a closed graph does not have perfect clique, then its Borel chromatic number is at most ℵ 1 in the Sacks model — they are all equally small. Are there two closed graphs without perfect cliques with uncountable but consistently different Borel chromatic numbers? 43 / 132
Review: definable graphs and definable chromatic numbers The G 0 graph and the G 0 -dichotomy What is the Borel chromatic number of G 0 ? Separating Borel chromatic number of closed graphs In the Cohen model: if the Borel chromatic number of an analytic graph is uncountable, then it has the cardinality of the continuum — they are all equally big. In the Sacks model: if a closed graph does not have perfect clique, then its Borel chromatic number is at most ℵ 1 in the Sacks model — they are all equally small. Are there two closed graphs without perfect cliques with uncountable but consistently different Borel chromatic numbers? We answer this question affirmatively. 44 / 132
Review: definable graphs and definable chromatic numbers The G 0 graph and the G 0 -dichotomy What is the Borel chromatic number of G 0 ? Separating Borel chromatic number of closed graphs Define the graph G 1 by ( x , y ) ∈ G 1 ↔ ∃ ! n ∈ ω ( x ( n ) � = y ( n )) . 45 / 132
Review: definable graphs and definable chromatic numbers The G 0 graph and the G 0 -dichotomy What is the Borel chromatic number of G 0 ? Separating Borel chromatic number of closed graphs Define the graph G 1 by ( x , y ) ∈ G 1 ↔ ∃ ! n ∈ ω ( x ( n ) � = y ( n )) . G 0 ⊆ G 1 and, just like G 0 , G 1 is a closed graph without perfect cliques. 46 / 132
Review: definable graphs and definable chromatic numbers The G 0 graph and the G 0 -dichotomy What is the Borel chromatic number of G 0 ? Separating Borel chromatic number of closed graphs Define the graph G 1 by ( x , y ) ∈ G 1 ↔ ∃ ! n ∈ ω ( x ( n ) � = y ( n )) . G 0 ⊆ G 1 and, just like G 0 , G 1 is a closed graph without perfect cliques. G 0 is a forest; G 1 has even cycles but does not have odd cycles, therefore is bipartite. This implies χ ( G 0 ) = χ ( G 1 ) = 2 47 / 132
Review: definable graphs and definable chromatic numbers The G 0 graph and the G 0 -dichotomy What is the Borel chromatic number of G 0 ? Separating Borel chromatic number of closed graphs Define the graph G 1 by ( x , y ) ∈ G 1 ↔ ∃ ! n ∈ ω ( x ( n ) � = y ( n )) . G 0 ⊆ G 1 and, just like G 0 , G 1 is a closed graph without perfect cliques. G 0 is a forest; G 1 has even cycles but does not have odd cycles, therefore is bipartite. This implies χ ( G 0 ) = χ ( G 1 ) = 2 χ B ( G 0 ) ≥ χ BP ( G 0 ) ≥ cov ( M ) 48 / 132
Review: definable graphs and definable chromatic numbers The G 0 graph and the G 0 -dichotomy What is the Borel chromatic number of G 0 ? Separating Borel chromatic number of closed graphs Define the graph G 1 by ( x , y ) ∈ G 1 ↔ ∃ ! n ∈ ω ( x ( n ) � = y ( n )) . G 0 ⊆ G 1 and, just like G 0 , G 1 is a closed graph without perfect cliques. G 0 is a forest; G 1 has even cycles but does not have odd cycles, therefore is bipartite. This implies χ ( G 0 ) = χ ( G 1 ) = 2 χ B ( G 0 ) ≥ χ BP ( G 0 ) ≥ cov ( M ) χ µ ( G 1 ) ≥ cov ( N ) 49 / 132
Review: definable graphs and definable chromatic numbers The G 0 graph and the G 0 -dichotomy What is the Borel chromatic number of G 0 ? Separating Borel chromatic number of closed graphs Define the graph G 1 by ( x , y ) ∈ G 1 ↔ ∃ ! n ∈ ω ( x ( n ) � = y ( n )) . G 0 ⊆ G 1 and, just like G 0 , G 1 is a closed graph without perfect cliques. G 0 is a forest; G 1 has even cycles but does not have odd cycles, therefore is bipartite. This implies χ ( G 0 ) = χ ( G 1 ) = 2 χ B ( G 0 ) ≥ χ BP ( G 0 ) ≥ cov ( M ) χ µ ( G 1 ) ≥ cov ( N ) χ µ ( G 0 ) = 3 (see [Mil08]). 50 / 132
Review: definable graphs and definable chromatic numbers The G 0 graph and the G 0 -dichotomy What is the Borel chromatic number of G 0 ? Separating Borel chromatic number of closed graphs From the previews this we get χ B ( G 1 ) ≥ max { cov ( N ) , cov ( M ) } . 51 / 132
Review: definable graphs and definable chromatic numbers The G 0 graph and the G 0 -dichotomy What is the Borel chromatic number of G 0 ? Separating Borel chromatic number of closed graphs From the previews this we get χ B ( G 1 ) ≥ max { cov ( N ) , cov ( M ) } . The fact that χ µ ( G 0 ) = 3 is a good indicative that we may be able to increase χ B ( G 1 ) without affecting χ B ( G 0 ) . 52 / 132
Review: definable graphs and definable chromatic numbers The G 0 graph and the G 0 -dichotomy What is the Borel chromatic number of G 0 ? Separating Borel chromatic number of closed graphs From the previews this we get χ B ( G 1 ) ≥ max { cov ( N ) , cov ( M ) } . The fact that χ µ ( G 0 ) = 3 is a good indicative that we may be able to increase χ B ( G 1 ) without affecting χ B ( G 0 ) . One idea would be to increase cov ( N ) and hope that keeping cov ( M ) small it will not increase χ B ( G 0 ) 53 / 132
Review: definable graphs and definable chromatic numbers The G 0 graph and the G 0 -dichotomy What is the Borel chromatic number of G 0 ? Separating Borel chromatic number of closed graphs From the previews this we get χ B ( G 1 ) ≥ max { cov ( N ) , cov ( M ) } . The fact that χ µ ( G 0 ) = 3 is a good indicative that we may be able to increase χ B ( G 1 ) without affecting χ B ( G 0 ) . One idea would be to increase cov ( N ) and hope that keeping cov ( M ) small it will not increase χ B ( G 0 ) — a good candidate for this is the random forcing. We proved that every random real is contained in a Borel G 0 -independet set coded in the ground model. 54 / 132
Review: definable graphs and definable chromatic numbers The G 0 graph and the G 0 -dichotomy What is the Borel chromatic number of G 0 ? Separating Borel chromatic number of closed graphs From the previews this we get χ B ( G 1 ) ≥ max { cov ( N ) , cov ( M ) } . The fact that χ µ ( G 0 ) = 3 is a good indicative that we may be able to increase χ B ( G 1 ) without affecting χ B ( G 0 ) . One idea would be to increase cov ( N ) and hope that keeping cov ( M ) small it will not increase χ B ( G 0 ) — a good candidate for this is the random forcing. We proved that every random real is contained in a Borel G 0 -independet set coded in the ground model. We do not know whether the same can be said about any other real added in the random extension. 55 / 132
Review: definable graphs and definable chromatic numbers The G 0 graph and the G 0 -dichotomy What is the Borel chromatic number of G 0 ? Separating Borel chromatic number of closed graphs Definition A tree p ⊆ 2 <ω is perfect iff for every t ∈ p there is s ∈ p such that t ⊆ s and s is a splitting node of p — i.e., s � 0 , s � 1 ∈ p . 56 / 132
Review: definable graphs and definable chromatic numbers The G 0 graph and the G 0 -dichotomy What is the Borel chromatic number of G 0 ? Separating Borel chromatic number of closed graphs Definition A tree p ⊆ 2 <ω is perfect iff for every t ∈ p there is s ∈ p such that t ⊆ s and s is a splitting node of p — i.e., s � 0 , s � 1 ∈ p . A perfect tree p is uniform, or a Silver tree, iff for all s , t ∈ p | s | = | t | → s ⌢ i ∈ p ↔ t ⌢ i ∈ p 57 / 132
Review: definable graphs and definable chromatic numbers The G 0 graph and the G 0 -dichotomy What is the Borel chromatic number of G 0 ? Separating Borel chromatic number of closed graphs Definition A tree p ⊆ 2 <ω is perfect iff for every t ∈ p there is s ∈ p such that t ⊆ s and s is a splitting node of p — i.e., s � 0 , s � 1 ∈ p . A perfect tree p is uniform, or a Silver tree, iff for all s , t ∈ p | s | = | t | → s ⌢ i ∈ p ↔ t ⌢ i ∈ p The Silver forcing V consists of uniform trees ordered by inclusion. 58 / 132
Review: definable graphs and definable chromatic numbers The G 0 graph and the G 0 -dichotomy What is the Borel chromatic number of G 0 ? Separating Borel chromatic number of closed graphs Definition A tree p ⊆ 2 <ω is perfect iff for every t ∈ p there is s ∈ p such that t ⊆ s and s is a splitting node of p — i.e., s � 0 , s � 1 ∈ p . A perfect tree p is uniform, or a Silver tree, iff for all s , t ∈ p | s | = | t | → s ⌢ i ∈ p ↔ t ⌢ i ∈ p The Silver forcing V consists of uniform trees ordered by inclusion. Remark It is the same as the forcing notion of parcial function from ω to 2 with co-infinite domain ordered by direct inclusion. 59 / 132
Review: definable graphs and definable chromatic numbers The G 0 graph and the G 0 -dichotomy What is the Borel chromatic number of G 0 ? Separating Borel chromatic number of closed graphs For i ∈ 2, let I 0 and I 1 be the σ -ideal Borel generated by countable unions of G 0 and G 1 -independent sets, respectively. 60 / 132
Review: definable graphs and definable chromatic numbers The G 0 graph and the G 0 -dichotomy What is the Borel chromatic number of G 0 ? Separating Borel chromatic number of closed graphs For i ∈ 2, let I 0 and I 1 be the σ -ideal Borel generated by countable unions of G 0 and G 1 -independent sets, respectively. Theorem (Zapletal [Zap04]) Let A ⊆ 2 ω be an analytic set. Then either A ∈ I G 1 or there is a Silver tree p such that [ p ] ⊆ A 61 / 132
Review: definable graphs and definable chromatic numbers The G 0 graph and the G 0 -dichotomy What is the Borel chromatic number of G 0 ? Separating Borel chromatic number of closed graphs For i ∈ 2, let I 0 and I 1 be the σ -ideal Borel generated by countable unions of G 0 and G 1 -independent sets, respectively. Theorem (Zapletal [Zap04]) Let A ⊆ 2 ω be an analytic set. Then either A ∈ I G 1 or there is a Silver tree p such that [ p ] ⊆ A This means that the fuction p �→ [ p ] is a dense embeding from the Silver frocing into B ( 2 ω ) \ I G 1 . 62 / 132
Review: definable graphs and definable chromatic numbers The G 0 graph and the G 0 -dichotomy What is the Borel chromatic number of G 0 ? Separating Borel chromatic number of closed graphs For i ∈ 2, let I 0 and I 1 be the σ -ideal Borel generated by countable unions of G 0 and G 1 -independent sets, respectively. Theorem (Zapletal [Zap04]) Let A ⊆ 2 ω be an analytic set. Then either A ∈ I G 1 or there is a Silver tree p such that [ p ] ⊆ A This means that the fuction p �→ [ p ] is a dense embeding from the Silver frocing into B ( 2 ω ) \ I G 1 . Now we know that the generic real avoids any Borel set in I G 1 coded in the ground model (in particular the G 1 -independents), therefore it increases χ B ( G 1 ) . 63 / 132
Review: definable graphs and definable chromatic numbers The G 0 graph and the G 0 -dichotomy What is the Borel chromatic number of G 0 ? Separating Borel chromatic number of closed graphs For i ∈ 2, let I 0 and I 1 be the σ -ideal Borel generated by countable unions of G 0 and G 1 -independent sets, respectively. Theorem (Zapletal [Zap04]) Let A ⊆ 2 ω be an analytic set. Then either A ∈ I G 1 or there is a Silver tree p such that [ p ] ⊆ A This means that the fuction p �→ [ p ] is a dense embeding from the Silver frocing into B ( 2 ω ) \ I G 1 . Now we know that the generic real avoids any Borel set in I G 1 coded in the ground model (in particular the G 1 -independents), therefore it increases χ B ( G 1 ) . Furthermore, it is the best forcing to increase χ B ( G 1 ) (this is due to Zapletal [Zap08a]). 64 / 132
Review: definable graphs and definable chromatic numbers The G 0 graph and the G 0 -dichotomy What is the Borel chromatic number of G 0 ? Separating Borel chromatic number of closed graphs Let p ⊆ 2 <ω be a perfect tree. For n ∈ ω we denote by split n ( p ) the set of all t ∈ p that are minimal in p with respect to ⊆ such that below t there are exactly n proper splitting nodes in p . 65 / 132
Review: definable graphs and definable chromatic numbers The G 0 graph and the G 0 -dichotomy What is the Borel chromatic number of G 0 ? Separating Borel chromatic number of closed graphs Let p ⊆ 2 <ω be a perfect tree. For n ∈ ω we denote by split n ( p ) the set of all t ∈ p that are minimal in p with respect to ⊆ such that below t there are exactly n proper splitting nodes in p . We define a sequence of parcial orders ( ≤ n ) n ∈ ω by p ≤ n q ↔ p ≤ q ∧ split n ( p ) = split n ( q ) . 66 / 132
Review: definable graphs and definable chromatic numbers The G 0 graph and the G 0 -dichotomy What is the Borel chromatic number of G 0 ? Separating Borel chromatic number of closed graphs Let p ⊆ 2 <ω be a perfect tree. For n ∈ ω we denote by split n ( p ) the set of all t ∈ p that are minimal in p with respect to ⊆ such that below t there are exactly n proper splitting nodes in p . We define a sequence of parcial orders ( ≤ n ) n ∈ ω by p ≤ n q ↔ p ≤ q ∧ split n ( p ) = split n ( q ) . We say that a sequence ( p n ) n ∈ ω of Sacks (Silver) conditions is a fusion sequence for the Sacks (respec. Silver )forcing iff · · · ≤ n + 1 p n + 1 ≤ n p n ≤ n − 1 · · · ≤ 0 p 0 67 / 132
Review: definable graphs and definable chromatic numbers The G 0 graph and the G 0 -dichotomy What is the Borel chromatic number of G 0 ? Separating Borel chromatic number of closed graphs Let p ⊆ 2 <ω be a perfect tree. For n ∈ ω we denote by split n ( p ) the set of all t ∈ p that are minimal in p with respect to ⊆ such that below t there are exactly n proper splitting nodes in p . We define a sequence of parcial orders ( ≤ n ) n ∈ ω by p ≤ n q ↔ p ≤ q ∧ split n ( p ) = split n ( q ) . We say that a sequence ( p n ) n ∈ ω of Sacks (Silver) conditions is a fusion sequence for the Sacks (respec. Silver )forcing iff · · · ≤ n + 1 p n + 1 ≤ n p n ≤ n − 1 · · · ≤ 0 p 0 It should be noted that if If ( p n ) n ∈ ω is a fusion sequence for the Sacks or Silver forcing then q . = � n ∈ ω p n is a Sacks or a Silver condition. 68 / 132
Review: definable graphs and definable chromatic numbers The G 0 graph and the G 0 -dichotomy What is the Borel chromatic number of G 0 ? Separating Borel chromatic number of closed graphs In the model obtained by adding ℵ 2 Silver reals over a model of CH, we have χ B ( G 1 ) = ℵ 2 = 2 ℵ 0 . 69 / 132
Review: definable graphs and definable chromatic numbers The G 0 graph and the G 0 -dichotomy What is the Borel chromatic number of G 0 ? Separating Borel chromatic number of closed graphs In the model obtained by adding ℵ 2 Silver reals over a model of CH, we have χ B ( G 1 ) = ℵ 2 = 2 ℵ 0 . Definition A forcing notion P does not add G 0 -independent closed sets if we can force every element of 2 ω to be in a G 0 -independent closed set coded in the ground model. 70 / 132
Review: definable graphs and definable chromatic numbers The G 0 graph and the G 0 -dichotomy What is the Borel chromatic number of G 0 ? Separating Borel chromatic number of closed graphs In the model obtained by adding ℵ 2 Silver reals over a model of CH, we have χ B ( G 1 ) = ℵ 2 = 2 ℵ 0 . Definition A forcing notion P does not add G 0 -independent closed sets if we can force every element of 2 ω to be in a G 0 -independent closed set coded in the ground model. Theorem If α is any ordinal, then the countable support iterated Silver forcing V α does not add G 0 -independent closed sets. 71 / 132
Review: definable graphs and definable chromatic numbers The G 0 graph and the G 0 -dichotomy What is the Borel chromatic number of G 0 ? Separating Borel chromatic number of closed graphs In the model obtained by adding ℵ 2 Silver reals over a model of CH, we have χ B ( G 1 ) = ℵ 2 = 2 ℵ 0 . Definition A forcing notion P does not add G 0 -independent closed sets if we can force every element of 2 ω to be in a G 0 -independent closed set coded in the ground model. Theorem If α is any ordinal, then the countable support iterated Silver forcing V α does not add G 0 -independent closed sets. In this way, if CH holds in the ground model, then in the generic extension obtained by forcing with V ω 2 : 72 / 132
Review: definable graphs and definable chromatic numbers The G 0 graph and the G 0 -dichotomy What is the Borel chromatic number of G 0 ? Separating Borel chromatic number of closed graphs In the model obtained by adding ℵ 2 Silver reals over a model of CH, we have χ B ( G 1 ) = ℵ 2 = 2 ℵ 0 . Definition A forcing notion P does not add G 0 -independent closed sets if we can force every element of 2 ω to be in a G 0 -independent closed set coded in the ground model. Theorem If α is any ordinal, then the countable support iterated Silver forcing V α does not add G 0 -independent closed sets. In this way, if CH holds in the ground model, then in the generic extension obtained by forcing with V ω 2 : ℵ 1 = χ B ( G 0 ) < χ B ( G 1 ) = ℵ 2 = 2 ℵ 0 . 73 / 132
Review: definable graphs and definable chromatic numbers The G 0 graph and the G 0 -dichotomy What is the Borel chromatic number of G 0 ? Separating Borel chromatic number of closed graphs We first will look at a close property to not adding G -independent sets, for certain graphs G . 74 / 132
Review: definable graphs and definable chromatic numbers The G 0 graph and the G 0 -dichotomy What is the Borel chromatic number of G 0 ? Separating Borel chromatic number of closed graphs We first will look at a close property to not adding G -independent sets, for certain graphs G . Definition A forcing notion P has the 2-localization property if we can force every element of ω ω to be in set of branches of some ground model binary tree. 75 / 132
Review: definable graphs and definable chromatic numbers The G 0 graph and the G 0 -dichotomy What is the Borel chromatic number of G 0 ? Separating Borel chromatic number of closed graphs We first will look at a close property to not adding G -independent sets, for certain graphs G . Definition A forcing notion P has the 2-localization property if we can force every element of ω ω to be in set of branches of some ground model binary tree. The 2-localization property implies the Sacks property and Sacks property = Laver property + ω ω -bounding. 76 / 132
Review: definable graphs and definable chromatic numbers The G 0 graph and the G 0 -dichotomy What is the Borel chromatic number of G 0 ? Separating Borel chromatic number of closed graphs We first will look at a close property to not adding G -independent sets, for certain graphs G . Definition A forcing notion P has the 2-localization property if we can force every element of ω ω to be in set of branches of some ground model binary tree. The 2-localization property implies the Sacks property and Sacks property = Laver property + ω ω -bounding. Laver property and ω ω -bounding are both preserved under countable supported iterations of proper forcing notions. 77 / 132
Review: definable graphs and definable chromatic numbers The G 0 graph and the G 0 -dichotomy What is the Borel chromatic number of G 0 ? Separating Borel chromatic number of closed graphs We first will look at a close property to not adding G -independent sets, for certain graphs G . Definition A forcing notion P has the 2-localization property if we can force every element of ω ω to be in set of branches of some ground model binary tree. The 2-localization property implies the Sacks property and Sacks property = Laver property + ω ω -bounding. Laver property and ω ω -bounding are both preserved under countable supported iterations of proper forcing notions. By Bartoszynsky’s: P has the Sacks property ⇒ � P cof ( N ) = | 2 ω ∩ V | . 78 / 132
Review: definable graphs and definable chromatic numbers The G 0 graph and the G 0 -dichotomy What is the Borel chromatic number of G 0 ? Separating Borel chromatic number of closed graphs We first will look at a close property to not adding G -independent sets, for certain graphs G . Definition A forcing notion P has the 2-localization property if we can force every element of ω ω to be in set of branches of some ground model binary tree. The 2-localization property implies the Sacks property and Sacks property = Laver property + ω ω -bounding. Laver property and ω ω -bounding are both preserved under countable supported iterations of proper forcing notions. By Bartoszynsky’s: P has the Sacks property ⇒ � P cof ( N ) = | 2 ω ∩ V | . if additionaly CH holds in V , then cof ( N ) = ℵ 1 79 / 132
Review: definable graphs and definable chromatic numbers The G 0 graph and the G 0 -dichotomy What is the Borel chromatic number of G 0 ? Separating Borel chromatic number of closed graphs We first will look at a close property to not adding G -independent sets, for certain graphs G . Definition A forcing notion P has the 2-localization property if we can force every element of ω ω to be in set of branches of some ground model binary tree. The 2-localization property implies the Sacks property and Sacks property = Laver property + ω ω -bounding. Laver property and ω ω -bounding are both preserved under countable supported iterations of proper forcing notions. By Bartoszynsky’s: P has the Sacks property ⇒ � P cof ( N ) = | 2 ω ∩ V | . if additionaly CH holds in V , then cof ( N ) = ℵ 1 80 / 132
Review: definable graphs and definable chromatic numbers The G 0 graph and the G 0 -dichotomy What is the Borel chromatic number of G 0 ? Separating Borel chromatic number of closed graphs Theorem If α is any ordinal, then the countable support iterated Silver forcing V α has the 2-localization property. 81 / 132
Review: definable graphs and definable chromatic numbers The G 0 graph and the G 0 -dichotomy What is the Borel chromatic number of G 0 ? Separating Borel chromatic number of closed graphs Theorem If α is any ordinal, then the countable support iterated Silver forcing V α has the 2-localization property. See [NR93], [Ros06], [RS08], [Zap08b] for more discussion. 82 / 132
Review: definable graphs and definable chromatic numbers The G 0 graph and the G 0 -dichotomy What is the Borel chromatic number of G 0 ? Separating Borel chromatic number of closed graphs Theorem If α is any ordinal, then the countable support iterated Silver forcing V α has the 2-localization property. See [NR93], [Ros06], [RS08], [Zap08b] for more discussion. x a P -name for an element of ω ω For P forcing notion and ˙ wtinessed by p , 83 / 132
Review: definable graphs and definable chromatic numbers The G 0 graph and the G 0 -dichotomy What is the Borel chromatic number of G 0 ? Separating Borel chromatic number of closed graphs Theorem If α is any ordinal, then the countable support iterated Silver forcing V α has the 2-localization property. See [NR93], [Ros06], [RS08], [Zap08b] for more discussion. x a P -name for an element of ω ω For P forcing notion and ˙ wtinessed by p , define for each q ≤ p , x ) = { s ∈ ω <ω | ∃ r ≤ q ( r � s ⊆ ˙ x ) } , T q ( ˙ the tree of q -possibilities for ˙ x . 84 / 132
Review: definable graphs and definable chromatic numbers The G 0 graph and the G 0 -dichotomy What is the Borel chromatic number of G 0 ? Separating Borel chromatic number of closed graphs Theorem If α is any ordinal, then the countable support iterated Silver forcing V α has the 2-localization property. See [NR93], [Ros06], [RS08], [Zap08b] for more discussion. x a P -name for an element of ω ω For P forcing notion and ˙ wtinessed by p , define for each q ≤ p , x ) = { s ∈ ω <ω | ∃ r ≤ q ( r � s ⊆ ˙ x ) } , T q ( ˙ the tree of q -possibilities for ˙ x . We have x ∈ [ T q ( ˙ q � ˙ x )] . 85 / 132
Review: definable graphs and definable chromatic numbers The G 0 graph and the G 0 -dichotomy What is the Borel chromatic number of G 0 ? Separating Borel chromatic number of closed graphs Each [ T q ( ˙ x )] is a closed set coded in the ground model that contains ˙ x . 86 / 132
Review: definable graphs and definable chromatic numbers The G 0 graph and the G 0 -dichotomy What is the Borel chromatic number of G 0 ? Separating Borel chromatic number of closed graphs Each [ T q ( ˙ x )] is a closed set coded in the ground model that contains ˙ x . Now for an arbitrary forcing notion, we can ask whether we can ensure that [ T q ( ˙ x )] has some desired property: binary? 87 / 132
Review: definable graphs and definable chromatic numbers The G 0 graph and the G 0 -dichotomy What is the Borel chromatic number of G 0 ? Separating Borel chromatic number of closed graphs Each [ T q ( ˙ x )] is a closed set coded in the ground model that contains ˙ x . Now for an arbitrary forcing notion, we can ask whether we can ensure that [ T q ( ˙ x )] has some desired property: binary? in I G 0 ? 88 / 132
Review: definable graphs and definable chromatic numbers The G 0 graph and the G 0 -dichotomy What is the Borel chromatic number of G 0 ? Separating Borel chromatic number of closed graphs Each [ T q ( ˙ x )] is a closed set coded in the ground model that contains ˙ x . Now for an arbitrary forcing notion, we can ask whether we can ensure that [ T q ( ˙ x )] has some desired property: binary? in I G 0 ? in I G 1 ? 89 / 132
Review: definable graphs and definable chromatic numbers The G 0 graph and the G 0 -dichotomy What is the Borel chromatic number of G 0 ? Separating Borel chromatic number of closed graphs Each [ T q ( ˙ x )] is a closed set coded in the ground model that contains ˙ x . Now for an arbitrary forcing notion, we can ask whether we can ensure that [ T q ( ˙ x )] has some desired property: binary? in I G 0 ? in I G 1 ? Lebesgue null? 90 / 132
Review: definable graphs and definable chromatic numbers The G 0 graph and the G 0 -dichotomy What is the Borel chromatic number of G 0 ? Separating Borel chromatic number of closed graphs Each [ T q ( ˙ x )] is a closed set coded in the ground model that contains ˙ x . Now for an arbitrary forcing notion, we can ask whether we can ensure that [ T q ( ˙ x )] has some desired property: binary? in I G 0 ? in I G 1 ? Lebesgue null? Let us first consider the case of adding one single Silver real. 91 / 132
Review: definable graphs and definable chromatic numbers The G 0 graph and the G 0 -dichotomy What is the Borel chromatic number of G 0 ? Separating Borel chromatic number of closed graphs Each [ T q ( ˙ x )] is a closed set coded in the ground model that contains ˙ x . Now for an arbitrary forcing notion, we can ask whether we can ensure that [ T q ( ˙ x )] has some desired property: binary? in I G 0 ? in I G 1 ? Lebesgue null? Let us first consider the case of adding one single Silver real. For a Silver tree q , consider the natural bijection between split n ( q ) and 2 n . 92 / 132
Review: definable graphs and definable chromatic numbers The G 0 graph and the G 0 -dichotomy What is the Borel chromatic number of G 0 ? Separating Borel chromatic number of closed graphs Each [ T q ( ˙ x )] is a closed set coded in the ground model that contains ˙ x . Now for an arbitrary forcing notion, we can ask whether we can ensure that [ T q ( ˙ x )] has some desired property: binary? in I G 0 ? in I G 1 ? Lebesgue null? Let us first consider the case of adding one single Silver real. For a Silver tree q , consider the natural bijection between split n ( q ) and 2 n . This induces for every σ ∈ 2 n , a corresponding q σ ∈ split n ( q ) , 93 / 132
Review: definable graphs and definable chromatic numbers The G 0 graph and the G 0 -dichotomy What is the Borel chromatic number of G 0 ? Separating Borel chromatic number of closed graphs Each [ T q ( ˙ x )] is a closed set coded in the ground model that contains ˙ x . Now for an arbitrary forcing notion, we can ask whether we can ensure that [ T q ( ˙ x )] has some desired property: binary? in I G 0 ? in I G 1 ? Lebesgue null? Let us first consider the case of adding one single Silver real. For a Silver tree q , consider the natural bijection between split n ( q ) and 2 n . This induces for every σ ∈ 2 n , a corresponding q σ ∈ split n ( q ) , then we define q ∗ σ = { s ∈ q | s ⊆ q σ ∨ q σ ⊆ s } ∈ V . 94 / 132
Review: definable graphs and definable chromatic numbers The G 0 graph and the G 0 -dichotomy What is the Borel chromatic number of G 0 ? Separating Borel chromatic number of closed graphs For a splitting level n , define the Silver tree � q ∗ n ⌢ i = q ∗ σ ⌢ i , σ ∈ 2 n for i ∈ 2. 95 / 132
Review: definable graphs and definable chromatic numbers The G 0 graph and the G 0 -dichotomy What is the Borel chromatic number of G 0 ? Separating Borel chromatic number of closed graphs For a splitting level n , define the Silver tree � q ∗ n ⌢ i = q ∗ σ ⌢ i , σ ∈ 2 n for i ∈ 2. A level n is ˙ x -indifferent for a condition q iff n is a splitting level of q and for every r ≤ q either 96 / 132
Review: definable graphs and definable chromatic numbers The G 0 graph and the G 0 -dichotomy What is the Borel chromatic number of G 0 ? Separating Borel chromatic number of closed graphs For a splitting level n , define the Silver tree � q ∗ n ⌢ i = q ∗ σ ⌢ i , σ ∈ 2 n for i ∈ 2. A level n is ˙ x -indifferent for a condition q iff n is a splitting level of q and for every r ≤ q either n is not a splitting level of q , 97 / 132
Review: definable graphs and definable chromatic numbers The G 0 graph and the G 0 -dichotomy What is the Borel chromatic number of G 0 ? Separating Borel chromatic number of closed graphs For a splitting level n , define the Silver tree � q ∗ n ⌢ i = q ∗ σ ⌢ i , σ ∈ 2 n for i ∈ 2. A level n is ˙ x -indifferent for a condition q iff n is a splitting level of q and for every r ≤ q either n is not a splitting level of q , or r ∗ n ⌢ 0 and r ∗ n ⌢ 1 are compatible about ˙ x . 98 / 132
Review: definable graphs and definable chromatic numbers The G 0 graph and the G 0 -dichotomy What is the Borel chromatic number of G 0 ? Separating Borel chromatic number of closed graphs For a splitting level n , define the Silver tree � q ∗ n ⌢ i = q ∗ σ ⌢ i , σ ∈ 2 n for i ∈ 2. A level n is ˙ x -indifferent for a condition q iff n is a splitting level of q and for every r ≤ q either n is not a splitting level of q , or r ∗ n ⌢ 0 and r ∗ n ⌢ 1 are compatible about ˙ x . We have the following dichotomy: (a) Either there is q ≤ p such that for all r ≤ q , no n is x -indifferent to r , or ˙ 99 / 132
Review: definable graphs and definable chromatic numbers The G 0 graph and the G 0 -dichotomy What is the Borel chromatic number of G 0 ? Separating Borel chromatic number of closed graphs For a splitting level n , define the Silver tree � q ∗ n ⌢ i = q ∗ σ ⌢ i , σ ∈ 2 n for i ∈ 2. A level n is ˙ x -indifferent for a condition q iff n is a splitting level of q and for every r ≤ q either n is not a splitting level of q , or r ∗ n ⌢ 0 and r ∗ n ⌢ 1 are compatible about ˙ x . We have the following dichotomy: (a) Either there is q ≤ p such that for all r ≤ q , no n is x -indifferent to r , or ˙ (b) for all q ≤ p there is r ≤ q and n that is ˙ x -indifferent to r . 100 / 132
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