Download A graph-theoretic approach to descriptive set theory and structural

A graph-theoretic approach to descriptive set
theory and structural dichotomy theorems
Benjamin Miller
Westfälische Wilhelms-Universität Münster
Lecture 3
June 30th , 2011
Part VI
Chromatic numbers
VI. Chromatic numbers
Basic definitions
Definition
A graph on X is an irreflexive symmetric set G ⊆ X × X .
Definition
The restriction of G to Y ⊆ X is the graph G Y on Y given by
G Y = G ∩ (Y × Y ).
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VI. Chromatic numbers
Basic definitions
Definition
We say that Y ⊆ X is G -dependent if G Y 6= ∅.
Definition
We say that Y ⊆ X is G -independent if it is not G -dependent.
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VI. Chromatic numbers
A canonical object
Definition
Fix sequences sn in 2n which are dense in the sense that
∀s ∈ 2<N ∃n ∈ N s v sn .
Definition
Let G0 denote the graph on 2N given by
G0 = {(sn a (i)a x, sn a (1 − i)a x) | i < 2, n ∈ N, and x ∈ 2N }.
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VI. Chromatic numbers
A canonical object
Proposition 1
Every G0 -independent set with the Baire property is meager.
Proof
Suppose that B ⊆ 2N has the Baire property but is not meager.
Fix s ∈ 2<N such that B is comeager in Ns .
Then there exists n ∈ N such that s v sn .
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VI. Chromatic numbers
A canonical object
Proof of Proposition 1 (continued)
It follows that the set C = B ∩ Nsn is comeager in Nsn .
Let ι be the homeomorphism of Nsn given by flipping the nth digit.
Then C and ι(C ) are comeager in Nsn , thus so too is C ∩ ι(C ).
Note that if x ∈ C ∩ ι(C ), then (x, ι(x)) ∈ G0 B.
It follows that B is G0 -dependent.
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VI. Chromatic numbers
A canonical object
Definition
A Y -coloring of G is a function c : X → Y such that
∀y ∈ Y c −1 ({y }) is G -independent.
Definition
A homomorphism from G ⊆ X × X to H ⊆ Y × Y is a function
ϕ : X → Y such that (ϕ × ϕ)(G ) ⊆ H.
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VI. Chromatic numbers
A canonical object
Proposition 2
There is no Borel ℵ0 -coloring of G0 .
Proof
Suppose that c : 2N → N is a Borel coloring of G0 .
Then each of the sets c −1 ({n}) is meager.
This contradicts the fact that 2N is a Baire space.
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VI. Chromatic numbers
A canonical object
Theorem 3 (Kechris-Solecki-Todorcevic)
Suppose that X is a Hausdorff space and G is an analytic graph on
X . Then exactly one of the following holds:
1
There is a Borel ℵ0 -coloring of G .
2
There is a continuous homomorphism from G0 to G .
We will prove this theorem in the final lecture.
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Part VII
Applications
VII. Applications
Co-analytic equivalence relations
Definition
The diagonal on X is the set ∆(X ) given by
∆(X ) = {(x, x) | x ∈ X }.
Definition
An embedding of an equivalence relation E on X into an equivalence
relation F on Y is an injection π : X → Y such that
∀x1 , x2 ∈ X (x1 E x2 ⇔ π(x1 ) F π(x2 )).
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VII. Applications
Co-analytic equivalence relations
Theorem 4 (Silver)
Suppose that X is a Hausdorff space and E is a co-analytic equivalence relation on X . Then exactly one of the following holds:
1
The equivalence relation E has only countably many classes.
2
There is a continuous embedding of ∆(2N ) into E .
Proof
Conditions (1) and (2) are clearly mutually exclusive.
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VII. Applications
Co-analytic equivalence relations
Proof of Theorem 4 (continued)
Define G = ∼E .
Note that every G -independent set is contained in a single E -class.
It follows that if there is an ℵ0 -coloring of G , then E has only
countably many equivalence classes.
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VII. Applications
Co-analytic equivalence relations
Proof of Theorem 4 (continued)
We can therefore assume that there is a continuous homomorphism
ϕ : 2N → X from G0 to G .
Define F = (ϕ × ϕ)−1 (E ).
Note that F is disjoint from G0 .
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VII. Applications
Co-analytic equivalence relations
Lemma 5
The equivalence relation F is meager.
Proof
As F is disjoint from G0 , every F -class is G0 -independent.
As F is co-analytic, so too is every F -class.
So every F -class has the Baire property, and is therefore meager.
It follows that F is meager.
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VII. Applications
Co-analytic equivalence relations
Proof of Theorem 4 (continued)
Fix a continuous embedding ψ of ∆(2N ) into F .
Then ϕ◦ψ is a continuous embedding of ∆(2N ) into E .
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VII. Applications
Co-analytic linear quasi-orders
Definition
We say that F is intersecting if ∀X , Y ∈ F X ∩ Y 6= ∅.
Definition
We say that C is a puncture set for F if ∀X ∈ F C ∩ X 6= ∅.
Definition
We say that F is separable if it has a countable puncture set.
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VII. Applications
Co-analytic linear quasi-orders
Proposition 6
Suppose that ≤ is a linear order for which there is an inseparable
intersecting family I of closed intervals. Then ≤ is not ccc.
Proof
Recursively find [xα , yα ] ∈ I such that ∀α < β < ω1 xα ∈
/ [xβ , yβ ].
As I is intersecting, it follows that ∀α < β < ω1 xα < xβ .
Then the intervals of the form (xλ , xλ+2 ), for limit ordinals λ < ω1 ,
are non-empty and pairwise disjoint.
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VII. Applications
Co-analytic linear quasi-orders
Definition
We say that a family F is σ-intersecting if it the union of countably
many intersecting families.
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VII. Applications
Co-analytic linear quasi-orders
Proposition 7
A linear order ≤ is separable if and only if the family I of closed
intervals with non-empty interior is σ-intersecting.
Proof
Suppose that ≤ is separable, and fix a countable dense set C .
Let Ix denote the family of closed intervals in I containing x.
Then each Ix is intersecting and I =
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S
x∈C
Ix .
VII. Applications
Co-analytic linear quasi-orders
Proof of Proposition 7 (continued)
Suppose now that I is σ-intersecting, and therefore ccc.
Fix intersecting families In such that I =
Fix puncture sets Cn for In .
Then the set C =
S
n∈N Cn
is dense.
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S
n∈N In .
VII. Applications
Co-analytic linear quasi-orders
Theorem 8 (Friedman-Shelah)
Suppose that X is a Hausdorff space and L is a linear co-analytic
quasi-order on X . Then L is separable if and only if L is ccc.
Proof
Clearly we can assume that ≡L has at least two equivalence classes.
It follows that X is the projection of ∼≡L onto either axis.
As ∼≡L is analytic, so too is X .
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VII. Applications
Co-analytic linear quasi-orders
Proof
Set I = {(x, y ) ∈ X × X | (x, y )L 6= ∅}.
Then I = {(x, y ) ∈ X × X | ∃z ∈ X (z, x), (y , z) ∈
/ L}.
It follows that I is analytic.
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VII. Applications
Co-analytic linear quasi-orders
Proof of Theorem 8 (continued)
Set G = {((x0 , y0 ), (x1 , y1 )) ∈ I × I | [x0 , y0 ]L ∩ [x1 , y1 ]L = ∅}.
Then G = {((x0 , y0 ), (x1 , y1 )) ∈ I × I | {(x0 , y1 ), (x1 , y0 )} * L}.
It follows that G is an analytic graph.
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VII. Applications
Co-analytic linear quasi-orders
Proof of Theorem 8 (continued)
Suppose first that there is an ℵ0 -coloring of G .
Then the set of closed L-intervals with non-empty interior is a σintersecting family.
It follows that L is separable.
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VII. Applications
Co-analytic linear quasi-orders
Proof of Theorem 8 (continued)
Suppose now that there is a continuous homomorphism ϕ : 2N → I
from G0 to G .
For each i, j < 2, define Rij ⊆ 2N × 2N by
Rij = {(x0 , x1 ) ∈ 2N × 2N | ϕi (xj ) ∈ [ϕ0 (x1−j ), ϕ1 (x1−j )]L }.
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VII. Applications
Co-analytic linear quasi-orders
Lemma 9
The relations Rij are meager.
Proof
We will only hand the case that i = j = 0.
As R00 is co-analytic, so too are each of its vertical sections.
In particular, each of its vertical sections has the Baire property.
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VII. Applications
Co-analytic linear quasi-orders
Proof of Lemma 9 (continued)
If (R00 )x is non-meager, then there exists (y , z) ∈ G0 (R00 )x .
So ϕ0 (x) ∈ [ϕ0 (y ), ϕ1 (y )]L ∩ [ϕ0 (z), ϕ1 (z)]L .
Thus (ϕ(y ), ϕ(z)) ∈
/ G , a contradiction.
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VII. Applications
Co-analytic linear quasi-orders
Proof of Theorem 8 (continued)
Set R = R00 ∪ R01 ∪ R10 ∪ R11 .
Fix a continuous embedding ψ of ∆(2N ) into R.
Then the intervals of the form ((ϕ0 ◦ ψ)(x), (ϕ1 ◦ ψ)(x))L have nonempty interior and are pairwise disjoint, so L is not ccc.
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