Two closed graphs with uncountable different Borel chromatic numbers - - PowerPoint PPT Presentation

Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

Two closed graphs with uncountable different Borel chromatic numbers

Michel Gaspar

Fachbereich Mathematik - Universität Hamburg michel.gaspar@uni-hamburg.de

27th May, 2019

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

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Review: definable graphs and definable chromatic numbers

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The G0 graph and the G0-dichotomy

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What is the Borel chromatic number of G0?

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Separating Borel chromatic number of closed graphs

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

Let X be any nonempty set and G be a graph on X (i.e., G is an irreflexive relation on X).

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

Let X be any nonempty set and G be a graph on X (i.e., G is an irreflexive relation on X). Recall that, for a cardinal κ, a κ-coloring of G is a function c : X → κ with the property that if (x, y) ∈ G then c(x) = c(y).

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

Let X be any nonempty set and G be a graph on X (i.e., G is an irreflexive relation on X). Recall that, for a cardinal κ, a κ-coloring of G is a function c : X → κ with the property that if (x, y) ∈ G then c(x) = c(y). The chromatic number of G is the least κ for which there exists a κ-coloring.

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

Let X be any nonempty set and G be a graph on X (i.e., G is an irreflexive relation on X). Recall that, for a cardinal κ, a κ-coloring of G is a function c : X → κ with the property that if (x, y) ∈ G then c(x) = c(y). The chromatic number of G is the least κ for which there exists a κ-coloring. Several important graphs are (irreflexive) relations on topological Hausdorff spaces (mostly on Polish spaces).

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

Let X be any nonempty set and G be a graph on X (i.e., G is an irreflexive relation on X). Recall that, for a cardinal κ, a κ-coloring of G is a function c : X → κ with the property that if (x, y) ∈ G then c(x) = c(y). The chromatic number of G is the least κ for which there exists a κ-coloring. Several important graphs are (irreflexive) relations on topological Hausdorff spaces (mostly on Polish spaces). This arises the question on what happens if we allow only definable types of colorings.

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

Let X be any nonempty set and G be a graph on X (i.e., G is an irreflexive relation on X). Recall that, for a cardinal κ, a κ-coloring of G is a function c : X → κ with the property that if (x, y) ∈ G then c(x) = c(y). The chromatic number of G is the least κ for which there exists a κ-coloring. Several important graphs are (irreflexive) relations on topological Hausdorff spaces (mostly on Polish spaces). This arises the question on what happens if we allow only definable types of colorings. What happens if we want to color the set of vertices in a graph in a definable way?

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

Let X be any nonempty set and G be a graph on X (i.e., G is an irreflexive relation on X). Recall that, for a cardinal κ, a κ-coloring of G is a function c : X → κ with the property that if (x, y) ∈ G then c(x) = c(y). The chromatic number of G is the least κ for which there exists a κ-coloring. Several important graphs are (irreflexive) relations on topological Hausdorff spaces (mostly on Polish spaces). This arises the question on what happens if we allow only definable types of colorings. What happens if we want to color the set of vertices in a graph in a definable way? How many colors do we need?

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

Definition Let X be a topological Hausdorff space and G be a graph on X. We say that G is closed (open, Borel, analytic etc) iff G is a closed (respec. open, Borel, analytic etc) subset of X 2 \ ∆(X).

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

Definition Let X be a topological Hausdorff space and G be a graph on X. We say that G is closed (open, Borel, analytic etc) iff G is a closed (respec. open, Borel, analytic etc) subset of X 2 \ ∆(X). For a family Γ of subsets of X, we say that a κ-coloring c of G is Γ-measurable iff c−1(α) ∈ Γ for every α < κ.

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

Definition Let X be a topological Hausdorff space and G be a graph on X. We say that G is closed (open, Borel, analytic etc) iff G is a closed (respec. open, Borel, analytic etc) subset of X 2 \ ∆(X). For a family Γ of subsets of X, we say that a κ-coloring c of G is Γ-measurable iff c−1(α) ∈ Γ for every α < κ. The Γ-chromatic number of G, denoted by χΓ(G), is the least κ for which there exists a κ-coloring of G which is Γ-measurable.

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

for Γ = P(X) we have the standard chromatic number.

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

for Γ = P(X) we have the standard chromatic number. for Γ = B(X) we have the Borel chromatic number, denoted by χB(G).

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

for Γ = P(X) we have the standard chromatic number. for Γ = B(X) we have the Borel chromatic number, denoted by χB(G).

• ther families Γ that might be interesting are the family of the

Lebesgue measurable sets or the family of sets with the Baire property.

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

for Γ = P(X) we have the standard chromatic number. for Γ = B(X) we have the Borel chromatic number, denoted by χB(G).

• ther families Γ that might be interesting are the family of the

Lebesgue measurable sets or the family of sets with the Baire property. See [Mil08] for more.

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

Definition Let (sn)n∈ω be a dense sequence of elements of 2<ω such that

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

Definition Let (sn)n∈ω be a dense sequence of elements of 2<ω such that |sn| = n for all n ∈ ω,

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

Definition Let (sn)n∈ω be a dense sequence of elements of 2<ω such that |sn| = n for all n ∈ ω, and every t ∈ 2<ω has an extension of the form sn.

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

Definition Let (sn)n∈ω be a dense sequence of elements of 2<ω such that |sn| = n for all n ∈ ω, and every t ∈ 2<ω has an extension of the form sn. The G0-graph is the graph on 2ω defined by G0 . = {(s

n 0x, s n 1x) | n ∈ ω ∧ x ∈ 2ω}.

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

Definition Let (sn)n∈ω be a dense sequence of elements of 2<ω such that |sn| = n for all n ∈ ω, and every t ∈ 2<ω has an extension of the form sn. The G0-graph is the graph on 2ω defined by G0 . = {(s

n 0x, s n 1x) | n ∈ ω ∧ x ∈ 2ω}.

Lemma The chromatic number of G0 is 2 and the Borel chromatic number

• f G0 is uncountable.

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

Definition Let (sn)n∈ω be a dense sequence of elements of 2<ω such that |sn| = n for all n ∈ ω, and every t ∈ 2<ω has an extension of the form sn. The G0-graph is the graph on 2ω defined by G0 . = {(s

n 0x, s n 1x) | n ∈ ω ∧ x ∈ 2ω}.

Lemma The chromatic number of G0 is 2 and the Borel chromatic number

• f G0 is uncountable. In fact, χB(G0) ≥ cov(M).

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

Definition Let (sn)n∈ω be a dense sequence of elements of 2<ω such that |sn| = n for all n ∈ ω, and every t ∈ 2<ω has an extension of the form sn. The G0-graph is the graph on 2ω defined by G0 . = {(s

n 0x, s n 1x) | n ∈ ω ∧ x ∈ 2ω}.

Lemma The chromatic number of G0 is 2 and the Borel chromatic number

• f G0 is uncountable. In fact, χB(G0) ≥ cov(M).

For proof see (see [KST99].

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

Theorem (Kechris-Solecki-Todorcevic G0-dichotomy, [KST99]) Let X be a Polish space and G be an analytic graph on X. Then exactly one of the following holds:

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

Theorem (Kechris-Solecki-Todorcevic G0-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 χB(G) ≤ ℵ0,

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

Theorem (Kechris-Solecki-Todorcevic G0-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 χB(G) ≤ ℵ0, or (b) there is a continuous homomorphism from G0 to G. In this case χB(G0) ≤ χB(G).

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

Theorem (Kechris-Solecki-Todorcevic G0-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 χB(G) ≤ ℵ0, or (b) there is a continuous homomorphism from G0 to G. In this case χB(G0) ≤ χB(G). The G0-dichotomy implies many known dichotomies in descritive set theory such as the perfect set property or the Silver’s dichotomy

• n co-analytic equivalence relations (see [Mil12]).

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

We want to understand the relationship between χB(G0) and the various cardinal invariants in the Cichón’s diagram.

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

We want to understand the relationship between χB(G0) and the various cardinal invariants in the Cichón’s diagram. First, notice that G has a perfect clique ⇒ χB(G) = 2ℵ0.

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

We want to understand the relationship between χB(G0) and the various cardinal invariants in the Cichón’s diagram. First, notice that G has a perfect clique ⇒ χB(G) = 2ℵ0. Now, G0 belongs to the class of closed graphs on the Cantor space.

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

We want to understand the relationship between χB(G0) and the various cardinal invariants in the Cichón’s diagram. First, notice that G has a perfect clique ⇒ χB(G) = 2ℵ0. Now, G0 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.

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

We want to understand the relationship between χB(G0) and the various cardinal invariants in the Cichón’s diagram. First, notice that G has a perfect clique ⇒ χB(G) = 2ℵ0. Now, G0 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.

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

First, in the model obtained by adding κ Cohen reals with finite support interation

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

First, in the model obtained by adding κ Cohen reals with finite support interation cov(M) = χB(G) = κ = 2ℵ0, for any analytic graph G on a Polish space with uncountable Borel chromatic number

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

First, in the model obtained by adding κ Cohen reals with finite support interation cov(M) = χB(G) = κ = 2ℵ0, 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) ≤ χB(G0) ≤ χB(G) ≤ 2ℵ0 for G as above).

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

First, in the model obtained by adding κ Cohen reals with finite support interation cov(M) = χB(G) = κ = 2ℵ0, 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) ≤ χB(G0) ≤ χB(G) ≤ 2ℵ0 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.

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? 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.

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? 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

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? 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 χB(G) ≤ ℵ1 in the Sacks model.

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? 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 χB(G) ≤ ℵ1 in the Sacks model. In this model, the continuum is 2ℵ0 = ℵ2. It remains open if the same holds for the classe of analytic graphs on Polish spaces.

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? 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.

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? 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 ℵ1 in the Sacks model — they are all equally small.

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? 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 ℵ1 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?

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? 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 ℵ1 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.

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

Define the graph G1 by (x, y) ∈ G1 ↔ ∃!n ∈ ω(x(n) = y(n)).

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

Define the graph G1 by (x, y) ∈ G1 ↔ ∃!n ∈ ω(x(n) = y(n)). G0 ⊆ G1 and, just like G0, G1 is a closed graph without perfect cliques.

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

Define the graph G1 by (x, y) ∈ G1 ↔ ∃!n ∈ ω(x(n) = y(n)). G0 ⊆ G1 and, just like G0, G1 is a closed graph without perfect cliques. G0 is a forest; G1 has even cycles but does not have odd cycles, therefore is bipartite. This implies χ(G0) = χ(G1) = 2

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

Define the graph G1 by (x, y) ∈ G1 ↔ ∃!n ∈ ω(x(n) = y(n)). G0 ⊆ G1 and, just like G0, G1 is a closed graph without perfect cliques. G0 is a forest; G1 has even cycles but does not have odd cycles, therefore is bipartite. This implies χ(G0) = χ(G1) = 2 χB(G0) ≥ χBP(G0) ≥ cov(M)

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

Define the graph G1 by (x, y) ∈ G1 ↔ ∃!n ∈ ω(x(n) = y(n)). G0 ⊆ G1 and, just like G0, G1 is a closed graph without perfect cliques. G0 is a forest; G1 has even cycles but does not have odd cycles, therefore is bipartite. This implies χ(G0) = χ(G1) = 2 χB(G0) ≥ χBP(G0) ≥ cov(M) χµ(G1) ≥ cov(N)

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

Define the graph G1 by (x, y) ∈ G1 ↔ ∃!n ∈ ω(x(n) = y(n)). G0 ⊆ G1 and, just like G0, G1 is a closed graph without perfect cliques. G0 is a forest; G1 has even cycles but does not have odd cycles, therefore is bipartite. This implies χ(G0) = χ(G1) = 2 χB(G0) ≥ χBP(G0) ≥ cov(M) χµ(G1) ≥ cov(N) χµ(G0) = 3 (see [Mil08]).

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

From the previews this we get χB(G1) ≥ max{cov(N), cov(M)}.

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

From the previews this we get χB(G1) ≥ max{cov(N), cov(M)}. The fact that χµ(G0) = 3 is a good indicative that we may be able to increase χB(G1) without affecting χB(G0).

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

From the previews this we get χB(G1) ≥ max{cov(N), cov(M)}. The fact that χµ(G0) = 3 is a good indicative that we may be able to increase χB(G1) without affecting χB(G0). One idea would be to increase cov(N) and hope that keeping cov(M) small it will not increase χB(G0)

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

From the previews this we get χB(G1) ≥ max{cov(N), cov(M)}. The fact that χµ(G0) = 3 is a good indicative that we may be able to increase χB(G1) without affecting χB(G0). One idea would be to increase cov(N) and hope that keeping cov(M) small it will not increase χB(G0) — a good candidate for this is the random forcing. We proved that every random real is contained in a Borel G0-independet set coded in the ground model.

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

From the previews this we get χB(G1) ≥ max{cov(N), cov(M)}. The fact that χµ(G0) = 3 is a good indicative that we may be able to increase χB(G1) without affecting χB(G0). One idea would be to increase cov(N) and hope that keeping cov(M) small it will not increase χB(G0) — a good candidate for this is the random forcing. We proved that every random real is contained in a Borel G0-independet set coded in the ground model. We do not know whether the same can be said about any

• ther real added in the random extension.

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? 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., s0, s1 ∈ p.

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? 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., s0, s1 ∈ 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

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? 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., s0, s1 ∈ 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.

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? 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., s0, s1 ∈ 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.

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

For i ∈ 2, let I0 and I1 be the σ-ideal Borel generated by countable unions of G0 and G1-independent sets, respectively.

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

For i ∈ 2, let I0 and I1 be the σ-ideal Borel generated by countable unions of G0 and G1-independent sets, respectively. Theorem (Zapletal [Zap04]) Let A ⊆ 2ω be an analytic set. Then either A ∈ IG1 or there is a Silver tree p such that [p] ⊆ A

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

For i ∈ 2, let I0 and I1 be the σ-ideal Borel generated by countable unions of G0 and G1-independent sets, respectively. Theorem (Zapletal [Zap04]) Let A ⊆ 2ω be an analytic set. Then either A ∈ IG1 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(2ω) \ IG1.

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

For i ∈ 2, let I0 and I1 be the σ-ideal Borel generated by countable unions of G0 and G1-independent sets, respectively. Theorem (Zapletal [Zap04]) Let A ⊆ 2ω be an analytic set. Then either A ∈ IG1 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(2ω) \ IG1. Now we know that the generic real avoids any Borel set in IG1 coded in the ground model (in particular the G1-independents), therefore it increases χB(G1).

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

For i ∈ 2, let I0 and I1 be the σ-ideal Borel generated by countable unions of G0 and G1-independent sets, respectively. Theorem (Zapletal [Zap04]) Let A ⊆ 2ω be an analytic set. Then either A ∈ IG1 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(2ω) \ IG1. Now we know that the generic real avoids any Borel set in IG1 coded in the ground model (in particular the G1-independents), therefore it increases χB(G1). Furthermore, it is the best forcing to increase χB(G1) (this is due to Zapletal [Zap08a]).

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

Let p ⊆ 2<ω be a perfect tree. For n ∈ ω we denote by splitn(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.

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

Let p ⊆ 2<ω be a perfect tree. For n ∈ ω we denote by splitn(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 ∧ splitn(p) = splitn(q).

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

Let p ⊆ 2<ω be a perfect tree. For n ∈ ω we denote by splitn(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 ∧ splitn(p) = splitn(q). We say that a sequence (pn)n∈ω of Sacks (Silver) conditions is a fusion sequence for the Sacks (respec. Silver )forcing iff · · · ≤n+1 pn+1 ≤n pn ≤n−1 · · · ≤0 p0

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

Let p ⊆ 2<ω be a perfect tree. For n ∈ ω we denote by splitn(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 ∧ splitn(p) = splitn(q). We say that a sequence (pn)n∈ω of Sacks (Silver) conditions is a fusion sequence for the Sacks (respec. Silver )forcing iff · · · ≤n+1 pn+1 ≤n pn ≤n−1 · · · ≤0 p0 It should be noted that if If (pn)n∈ω is a fusion sequence for the Sacks or Silver forcing then q . =

n∈ω pn is a Sacks or a Silver

condition.

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

In the model obtained by adding ℵ2 Silver reals over a model of CH, we have χB(G1) = ℵ2 = 2ℵ0.

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

In the model obtained by adding ℵ2 Silver reals over a model of CH, we have χB(G1) = ℵ2 = 2ℵ0. Definition A forcing notion P does not add G0-independent closed sets if we can force every element of 2ω to be in a G0-independent closed set coded in the ground model.

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

In the model obtained by adding ℵ2 Silver reals over a model of CH, we have χB(G1) = ℵ2 = 2ℵ0. Definition A forcing notion P does not add G0-independent closed sets if we can force every element of 2ω to be in a G0-independent closed set coded in the ground model. Theorem If α is any ordinal, then the countable support iterated Silver forcing Vα does not add G0-independent closed sets.

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

In the model obtained by adding ℵ2 Silver reals over a model of CH, we have χB(G1) = ℵ2 = 2ℵ0. Definition A forcing notion P does not add G0-independent closed sets if we can force every element of 2ω to be in a G0-independent closed set coded in the ground model. Theorem If α is any ordinal, then the countable support iterated Silver forcing Vα does not add G0-independent closed sets. In this way, if CH holds in the ground model, then in the generic extension obtained by forcing with Vω2:

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

In the model obtained by adding ℵ2 Silver reals over a model of CH, we have χB(G1) = ℵ2 = 2ℵ0. Definition A forcing notion P does not add G0-independent closed sets if we can force every element of 2ω to be in a G0-independent closed set coded in the ground model. Theorem If α is any ordinal, then the countable support iterated Silver forcing Vα does not add G0-independent closed sets. In this way, if CH holds in the ground model, then in the generic extension obtained by forcing with Vω2: ℵ1 = χB(G0) < χB(G1) = ℵ2 = 2ℵ0.

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? 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.

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? 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 ωω to be in set of branches of some ground model binary tree.

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? 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 ωω 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 + ωω-bounding.

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? 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 ωω 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 + ωω-bounding. Laver property and ωω-bounding are both preserved under countable supported iterations of proper forcing notions.

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? 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 ωω 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 + ωω-bounding. Laver property and ωω-bounding are both preserved under countable supported iterations of proper forcing notions. By Bartoszynsky’s: P has the Sacks property ⇒ P cof(N) = |2ω ∩ V |.

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? 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 ωω 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 + ωω-bounding. Laver property and ωω-bounding are both preserved under countable supported iterations of proper forcing notions. By Bartoszynsky’s: P has the Sacks property ⇒ P cof(N) = |2ω ∩ V |. if additionaly CH holds in V , then cof(N) = ℵ1

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? 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 ωω 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 + ωω-bounding. Laver property and ωω-bounding are both preserved under countable supported iterations of proper forcing notions. By Bartoszynsky’s: P has the Sacks property ⇒ P cof(N) = |2ω ∩ V |. if additionaly CH holds in V , then cof(N) = ℵ1

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

Theorem If α is any ordinal, then the countable support iterated Silver forcing Vα has the 2-localization property.

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

Theorem If α is any ordinal, then the countable support iterated Silver forcing Vα has the 2-localization property. See [NR93], [Ros06], [RS08], [Zap08b] for more discussion.

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

Theorem If α is any ordinal, then the countable support iterated Silver forcing Vα has the 2-localization property. See [NR93], [Ros06], [RS08], [Zap08b] for more discussion. For P forcing notion and ˙ x a P-name for an element of ωω wtinessed by p,

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

Theorem If α is any ordinal, then the countable support iterated Silver forcing Vα has the 2-localization property. See [NR93], [Ros06], [RS08], [Zap08b] for more discussion. For P forcing notion and ˙ x a P-name for an element of ωω wtinessed by p, define for each q ≤ p, Tq( ˙ x) = {s ∈ ω<ω | ∃r ≤ q(r s ⊆ ˙ x)}, the tree of q-possibilities for ˙ x.

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

Theorem If α is any ordinal, then the countable support iterated Silver forcing Vα has the 2-localization property. See [NR93], [Ros06], [RS08], [Zap08b] for more discussion. For P forcing notion and ˙ x a P-name for an element of ωω wtinessed by p, define for each q ≤ p, Tq( ˙ x) = {s ∈ ω<ω | ∃r ≤ q(r s ⊆ ˙ x)}, the tree of q-possibilities for ˙

• x. We have

q ˙ x ∈ [Tq( ˙ x)].

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

Each [Tq( ˙ x)] is a closed set coded in the ground model that contains ˙ x.

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

Each [Tq( ˙ 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 [Tq( ˙ x)] has some desired property: binary?

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

Each [Tq( ˙ 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 [Tq( ˙ x)] has some desired property: binary? in IG0?

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

Each [Tq( ˙ 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 [Tq( ˙ x)] has some desired property: binary? in IG0? in IG1?

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

Each [Tq( ˙ 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 [Tq( ˙ x)] has some desired property: binary? in IG0? in IG1? Lebesgue null?

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

Each [Tq( ˙ 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 [Tq( ˙ x)] has some desired property: binary? in IG0? in IG1? Lebesgue null? Let us first consider the case of adding one single Silver real.

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

Each [Tq( ˙ 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 [Tq( ˙ x)] has some desired property: binary? in IG0? in IG1? Lebesgue null? Let us first consider the case of adding one single Silver real. For a Silver tree q, consider the natural bijection between splitn(q) and 2n.

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

Each [Tq( ˙ 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 [Tq( ˙ x)] has some desired property: binary? in IG0? in IG1? Lebesgue null? Let us first consider the case of adding one single Silver real. For a Silver tree q, consider the natural bijection between splitn(q) and

• 2n. This induces for every σ ∈ 2n, a corresponding qσ ∈ splitn(q),

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

Each [Tq( ˙ 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 [Tq( ˙ x)] has some desired property: binary? in IG0? in IG1? Lebesgue null? Let us first consider the case of adding one single Silver real. For a Silver tree q, consider the natural bijection between splitn(q) and

• 2n. This induces for every σ ∈ 2n, a corresponding qσ ∈ splitn(q),

then we define q ∗ σ = {s ∈ q | s ⊆ qσ ∨ qσ ⊆ s} ∈ V.

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

For a splitting level n, define the Silver tree q ∗ n⌢i =

• σ∈2n

q ∗ σ⌢i, for i ∈ 2.

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

For a splitting level n, define the Silver tree q ∗ n⌢i =

• σ∈2n

q ∗ σ⌢i, 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

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

For a splitting level n, define the Silver tree q ∗ n⌢i =

• σ∈2n

q ∗ σ⌢i, 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,

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

For a splitting level n, define the Silver tree q ∗ n⌢i =

• σ∈2n

q ∗ σ⌢i, 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.

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

For a splitting level n, define the Silver tree q ∗ n⌢i =

• σ∈2n

q ∗ σ⌢i, 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

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

For a splitting level n, define the Silver tree q ∗ n⌢i =

• σ∈2n

q ∗ σ⌢i, 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.

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

For a splitting level n, define the Silver tree q ∗ n⌢i =

• σ∈2n

q ∗ σ⌢i, 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.

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

If condition (b) is satisfied, ˙ x is actually a name for a ground-model real.

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

If condition (b) is satisfied, ˙ x is actually a name for a ground-model

• real. If condition (a) is satisfied, we can make [Tq( ˙

x)] binary and we have a ground model name for a homeomorphism ˙ h : [q] → [Tq( ˙ x)] mapping the generic real onto ˙ x.

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

If condition (b) is satisfied, ˙ x is actually a name for a ground-model

• real. If condition (a) is satisfied, we can make [Tq( ˙

x)] binary and we have a ground model name for a homeomorphism ˙ h : [q] → [Tq( ˙ x)] mapping the generic real onto ˙ x. This helps us to handle the successor case for the property of not adding G0-independent closed sets.

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

If condition (b) is satisfied, ˙ x is actually a name for a ground-model

• real. If condition (a) is satisfied, we can make [Tq( ˙

x)] binary and we have a ground model name for a homeomorphism ˙ h : [q] → [Tq( ˙ x)] mapping the generic real onto ˙ x. This helps us to handle the successor case for the property of not adding G0-independent closed sets. We need to import the fusion technology to the iterations:

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

If condition (b) is satisfied, ˙ x is actually a name for a ground-model

• real. If condition (a) is satisfied, we can make [Tq( ˙

x)] binary and we have a ground model name for a homeomorphism ˙ h : [q] → [Tq( ˙ x)] mapping the generic real onto ˙ x. This helps us to handle the successor case for the property of not adding G0-independent closed sets. We need to import the fusion technology to the iterations: Let F be a finite subset of α and η : F → ω.

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

If condition (b) is satisfied, ˙ x is actually a name for a ground-model

• real. If condition (a) is satisfied, we can make [Tq( ˙

x)] binary and we have a ground model name for a homeomorphism ˙ h : [q] → [Tq( ˙ x)] mapping the generic real onto ˙ x. This helps us to handle the successor case for the property of not adding G0-independent closed sets. We need to import the fusion technology to the iterations: Let F be a finite subset of α and η : F → ω. For p, q ∈ Vα let p ≤F,η q ↔ ∀γ ∈ F(p ↾ γ p(γ) ≤η(γ) q(γ)).

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

A sequence (pn)n∈ω is a fusion sequence if there are an increasing sequence (Fn)n∈ω of finite subsets of α and a sequence (ηn : Fn → ω | n ∈ ω) satisfying that for all n ∈ ω

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

A sequence (pn)n∈ω is a fusion sequence if there are an increasing sequence (Fn)n∈ω of finite subsets of α and a sequence (ηn : Fn → ω | n ∈ ω) satisfying that for all n ∈ ω (a) pn+1 ≤Fn,ηn pn

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

A sequence (pn)n∈ω is a fusion sequence if there are an increasing sequence (Fn)n∈ω of finite subsets of α and a sequence (ηn : Fn → ω | n ∈ ω) satisfying that for all n ∈ ω (a) pn+1 ≤Fn,ηn pn (b) for all γ ∈ Fn we have ηn(γ) ≤ ηn+1(γ)

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

A sequence (pn)n∈ω is a fusion sequence if there are an increasing sequence (Fn)n∈ω of finite subsets of α and a sequence (ηn : Fn → ω | n ∈ ω) satisfying that for all n ∈ ω (a) pn+1 ≤Fn,ηn pn (b) for all γ ∈ Fn we have ηn(γ) ≤ ηn+1(γ) (c) for all γ ∈ supt(pn) there is m ∈ ω such that γ ∈ Fm and ηm(γ) ≥ n

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

A sequence (pn)n∈ω is a fusion sequence if there are an increasing sequence (Fn)n∈ω of finite subsets of α and a sequence (ηn : Fn → ω | n ∈ ω) satisfying that for all n ∈ ω (a) pn+1 ≤Fn,ηn pn (b) for all γ ∈ Fn we have ηn(γ) ≤ ηn+1(γ) (c) for all γ ∈ supt(pn) there is m ∈ ω such that γ ∈ Fm and ηm(γ) ≥ n The fusion pω of a fusion sequence (pn)n∈ω in Vα is defined recursively by pω ↾ γ pω(γ) =

• n∈ω

pn(γ).

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

Let p ∈ Vα, F a finite subset of α and η : F → ω. For σ ∈

γ∈F 2η(γ) we define a condition p ∗ σ ∈ Vα by

(p ∗ σ) ↾ δ (p ∗ σ)(δ) = p(δ) ∗ σ(δ) if δ ∈ F, and (p ∗ σ)(δ) = p(δ) if δ ∈ α \ F. For F a finite subset of α and η : F → ω, the right notion of faithfulness for a condition p ∈ Vα can give provide us with various preservation theorems in the Silver model.

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

For F, η as above (a) we can say that p ∈ Vα is (F, η)-faithfull iff for every σ, τ ∈

γ∈F 2η(γ) such that σ = τ, ˙

xp∗σ and ˙ xp∗τ — the maximal initial segments of ˙ x decided by p ∗ σ and p ∗ τ, respectively — are incompatible.

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

For F, η as above (a) we can say that p ∈ Vα is (F, η)-faithfull iff for every σ, τ ∈

γ∈F 2η(γ) such that σ = τ, ˙

xp∗σ and ˙ xp∗τ — the maximal initial segments of ˙ x decided by p ∗ σ and p ∗ τ, respectively — are incompatible. (b) we can say that p ∈ Vα is (F, η)-faithfull iff for every σ, τ ∈

γ∈F 2η(γ) such that σ = τ, [Tp∗σ( ˙

x)] and [Tp∗τ( ˙ x)] are relatively G0-independent — i.e., there is no element of [Tp∗σ( ˙ x)] forming a G0-edge with some element of [Tp∗τ( ˙ x)]

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

For F, η as above (a) we can say that p ∈ Vα is (F, η)-faithfull iff for every σ, τ ∈

γ∈F 2η(γ) such that σ = τ, ˙

xp∗σ and ˙ xp∗τ — the maximal initial segments of ˙ x decided by p ∗ σ and p ∗ τ, respectively — are incompatible. (b) we can say that p ∈ Vα is (F, η)-faithfull iff for every σ, τ ∈

γ∈F 2η(γ) such that σ = τ, [Tp∗σ( ˙

x)] and [Tp∗τ( ˙ x)] are relatively G0-independent — i.e., there is no element of [Tp∗σ( ˙ x)] forming a G0-edge with some element of [Tp∗τ( ˙ x)] In the first case we get the preservation of the 2-localization property.

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

For F, η as above (a) we can say that p ∈ Vα is (F, η)-faithfull iff for every σ, τ ∈

γ∈F 2η(γ) such that σ = τ, ˙

xp∗σ and ˙ xp∗τ — the maximal initial segments of ˙ x decided by p ∗ σ and p ∗ τ, respectively — are incompatible. (b) we can say that p ∈ Vα is (F, η)-faithfull iff for every σ, τ ∈

γ∈F 2η(γ) such that σ = τ, [Tp∗σ( ˙

x)] and [Tp∗τ( ˙ x)] are relatively G0-independent — i.e., there is no element of [Tp∗σ( ˙ x)] forming a G0-edge with some element of [Tp∗τ( ˙ x)] In the first case we get the preservation of the 2-localization

• property. In the second case we get the preservation of not adding

G0-independent closed sets.

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

Lemma If ˙ x is a V-name for an element of 2ω witnessed by p, then

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

Lemma If ˙ x is a V-name for an element of 2ω witnessed by p, then (a) either there is q ≤ p such that [Tq( ˙ x)] is a Silver tree and, furthermore, we may assume [Tq( ˙ x)] is a G0-independent set,

• r

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

Lemma If ˙ x is a V-name for an element of 2ω witnessed by p, then (a) either there is q ≤ p such that [Tq( ˙ x)] is a Silver tree and, furthermore, we may assume [Tq( ˙ x)] is a G0-independent set,

• r

(b) there is q such that [Tq( ˙ x)] is a G1-independent set (hence G0-independent).

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

Lemma If ˙ x is a V-name for an element of 2ω witnessed by p, then (a) either there is q ≤ p such that [Tq( ˙ x)] is a Silver tree and, furthermore, we may assume [Tq( ˙ x)] is a G0-independent set,

• r

(b) there is q such that [Tq( ˙ x)] is a G1-independent set (hence G0-independent). For the part (a), it is useful to note that for any Silver tree p, there is q ≤ p such that [q] is G0-independent.

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

Lemma If ˙ x is a V-name for an element of 2ω witnessed by p, then (a) either there is q ≤ p such that [Tq( ˙ x)] is a Silver tree and, furthermore, we may assume [Tq( ˙ x)] is a G0-independent set,

• r

(b) there is q such that [Tq( ˙ x)] is a G1-independent set (hence G0-independent). For the part (a), it is useful to note that for any Silver tree p, there is q ≤ p such that [q] is G0-independent. This means that the Silver real is always in some closed G0-independent ground model set.

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

Theorem Let α be an ordinal, Vα+1 an α + 1 countable supported iteration

• f copies of the Silver forcing and and let ˙

x be a Vα+1-name for an element of 2ω only added at stage α + 1 and p ∈ Vα+1 such that p ˙ x : ω → 2 witnessing that.

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

Theorem Let α be an ordinal, Vα+1 an α + 1 countable supported iteration

• f copies of the Silver forcing and and let ˙

x be a Vα+1-name for an element of 2ω only added at stage α + 1 and p ∈ Vα+1 such that p ˙ x : ω → 2 witnessing that. Then there is q ≤ p such that [Tq( ˙ x)] is a G0-independent set.

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

Using the previows lemma and the maximal principle, there are Vα-names ˙ q for a condition and ˙ h a homeomorphism such that p ↾ α forces ˙ q ≤ p(α), ˙ h : ˙ q → [T ˙

q( ˙

x)] maps the generic real onto ˙ x, and either

(a) T ˙

q( ˙

x) is a Silver tree such that [T ˙

q( ˙

x)] is G0-independent, or (b) any two incompatible nodes of T ˙

q( ˙

x) disagree in at least two coordinates.

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

Theorem Let α be a limit ordinal, Vα an α countable supported iteration of copies of the Silver forcing and let ˙ x be a Vα-name for an element

• f 2ω only added at stage α and p ∈ Vα such that p ˙

x : ω → 2 witnessing that.

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

Theorem Let α be a limit ordinal, Vα an α countable supported iteration of copies of the Silver forcing and let ˙ x be a Vα-name for an element

• f 2ω only added at stage α and p ∈ Vα such that p ˙

x : ω → 2 witnessing that. Then there is q ≤ p such that [Tq( ˙ x)] is a G1-independent set

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

Theorem Let α be a limit ordinal, Vα an α countable supported iteration of copies of the Silver forcing and let ˙ x be a Vα-name for an element

• f 2ω only added at stage α and p ∈ Vα such that p ˙

x : ω → 2 witnessing that. Then there is q ≤ p such that [Tq( ˙ x)] is a G1-independent set (hence G0-independent).

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

Theorem Let α be a limit ordinal, Vα an α countable supported iteration of copies of the Silver forcing and let ˙ x be a Vα-name for an element

• f 2ω only added at stage α and p ∈ Vα such that p ˙

x : ω → 2 witnessing that. Then there is q ≤ p such that [Tq( ˙ x)] is a G1-independent set (hence G0-independent). In particular, no Silver reals are added at limit steps of countable supported iterations of the Silver forcing.

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

Alexander S Kechris, Slawomir Solecki, and Stevo Todorcevic. Borel chromatic numbers. Advances in Mathematics, 141(1):1–44, 1999. Benjamin D Miller. Measurable chromatic numbers. The Journal of Symbolic Logic, 73(4):1139–1157, 2008. Benjamin D Miller. The graph-theoretic approach to descriptive set theory. The Bulletin of Symbolic Logic, pages 554–575, 2012. Ludomir Newelski and Andrzej Rosłanowski. The ideal determined by the unsymmetric game. Proceedings of the American Mathematical Society, 117(3):823–831, 1993.

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Review: definable graphs and definable chromatic numbers The G0 graph and the G0-dichotomy What is the Borel chromatic number of G0? Separating Borel chromatic number of closed graphs

Andrzej Rosłanowski. n–localization property. The Journal of Symbolic Logic, 71(3):881–902, 2006. Andrzej Rosłanowski and Juris Stepr¯ ans. Chasing silver. Canadian Mathematical Bulletin, 51(4):593–603, 2008. Jindřich Zapletal. Descriptive set theory and definable forcing. American Mathematical Soc., 2004. Jindřich Zapletal. Forcing idealized, volume 174. Cambridge University Press Cambridge, 2008.

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Jindrich Zapletal. n-localization property in iterations. 2008.

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