Definable Graphs and Dominating Reals

Transcription

1 May 13, 2016

2 Graphs A graph on X is a symmetric, irreflexive G X 2. A graph G on a Polish space X is Borel if it is a Borel subset of X 2 \ {(x, x) : x X }. A X is an anticlique (G-free, independent) if for all x, y A we have (x, y) / G. C X is a clique if for all distinct x, y C we have (x, y) G.

3 Theorem (Geschke) Let G be a closed graph on ω ω without perfect cliques. Then there is a ccc forcing extension V [H] such that (ω ω ) V is covered by countably many compact G-anticliques.

4 Theorem (Geschke) Let G be a closed graph on a Polish space X. Then either G has a perfect clique or there is a ccc forcing extension where ℵ 1 < c and X is covered by ℵ 1 -many compact G-anticliques.

5 Dominating Reals In ω ω say x y if x(n) y(n) for all but finitely many n. Say V [H] has a dominating real if there is some y (ω ω ) V [H] such that x y for all x (ω ω ) V.

6 Dominating Reals Topological Rephrasing: In ω ω, sets of the form {x : x y} are K σ and every K σ set is contained in such a set.

7 Dominating Reals Topological Rephrasing: In ω ω, sets of the form {x : x y} are K σ and every K σ set is contained in such a set. So adding a dominating real is equivalent to covering (ω ω ) V by a K σ set.

8 Dominating Reals Question For a graph G on a Polish space X, when can we cover X V by countably many compact G-anticliques without adding a dominating real?

9 Adding Dominating Reals? If X contains a closed copy of ω ω, we must add a dominating real.

10 Adding Dominating Reals? If X contains a closed copy of ω ω, we must add a dominating real. Theorem (Hurewicz) For X Polish, X has a closed subspace homeomorphic to ω ω iff X isn t K σ.

11 Adding Dominating Reals. Define the F σ graph D on 2 ω by xdy if x has finitely many 1 s and y agrees with x up to its last 1 (or vice versa)

12 Adding Dominating Reals. Define the F σ graph D on 2 ω by xdy if x has finitely many 1 s and y agrees with x up to its last 1 (or vice versa) Let A 2 ω be the sequences with finitely many ones and B = A c. If C 2 ω is a closed D-anticlique, then C B is closed in 2 ω.

13 Adding Dominating Reals. Define the F σ graph D on 2 ω by xdy if x has finitely many 1 s and y agrees with x up to its last 1 (or vice versa) Let A 2 ω be the sequences with finitely many ones and B = A c. If C 2 ω is a closed D-anticlique, then C B is closed in 2 ω. By covering 2 ω by countably many closed D-anticliques, 2 ω = C n, we also have B = (B C n ), so B is K σ. But B is homeomorphic to ω ω.

14 Loose Graphs Definition Let G be a graph on a Polish space X. Say B X is G-loose if there is no {x n } B such that x n x and x n Gx for all n ω. Say G is loose if X = B n where each B n is G-loose.

15 Loose Graphs Theorem Let G be a closed, loose graph on a K σ Polish space X. Then there is a ccc forcing extension with no dominating reals such that the ground model elements of X are covered by countably many compact G-anticliques.

16 Loose Graphs Question Which graphs are loose?

17 Closure Properties If G on X is loose and F G, then F is loose. If G 1, G 2 are loose graphs on X, Y respectively, then G 1 G 2 is loose on X Y. If G on X is loose, H is a graph on Y and f : Y X is a continuous homomorphism, then H is loose.

18 Nonexamples Claim The graph D on 2 ω where xdy if x has finitely many 1 s and y agrees with x up to its last 1 (or vice versa) is not loose Suppose 2 ω = B n where the B n are all D-loose. We may assume the B n are contained in B or are a singleton containing one element of A. For the B n B, we know B n A =. So B = B n, hence B is K σ, a contradiction.

19 Nonexamples Proposition If G on a K σ space X has a perfect clique, then G is not loose.

20 Nonexamples Proposition If G on a K σ space X has a perfect clique, then G is not loose. Proof. Write X = K m for compact K m and let X = B n.

21 Nonexamples Proposition If G on a K σ space X has a perfect clique, then G is not loose. Proof. Write X = K m for compact K m and let X = B n. For some m, n ω, B n K m must contain infinitely many elements of the clique. Let x be a limit point of these elements. This shows B n isn t G-loose since x is also in the clique.

22 Examples Proposition If G on X is locally countable, then G is loose.

23 Examples Proposition If G on X is locally countable, then G is loose. Proof. If G is locally countable, each connected component is countable. Write X = B n where each B n contains at most one element from each component. Then each B n is G-loose since any element of X can share an edge with at most of element of B n.

24 Examples Acyclic graphs are loose. For functions f i : X X, graphs of the form G = G f1,...,f n are loose (where xgy if f i (x) = y or f i (y) = x for some i).

25 Examples Acyclic graphs are loose. For functions f i : X X, graphs of the form G = G f1,...,f n are loose (where xgy if f i (x) = y or f i (y) = x for some i). More generally, if G can be oriented so each vertex has finite out-degree, then G is loose.

26 Definable Looseness What if we require the loose sets to be Borel? Definition Say that G on X is Borel loose if G is loose, witnessed by X = B n for Borel sets B n.

27 G 0 There is a graph G 0 on 2 ω which is closed, locally countable, acyclic, has uncountable Borel chromatic number (since every nonmeager Borel set has a G 0 edge), and the following theorem holds:

28 G 0 There is a graph G 0 on 2 ω which is closed, locally countable, acyclic, has uncountable Borel chromatic number (since every nonmeager Borel set has a G 0 edge), and the following theorem holds: Theorem (Kechris-Solecki-Todorcevic) For an analytic graph G, exactly one of the following holds: G has countable Borel chromatic number. There is a continuous homomorphism of G 0 to G.

29 Borel-looseness Theorem A nonmeager B 2 ω with the Baire Property isn t G 0 -loose. In particular, G 0 isn t Borel-loose.

30 Borel-looseness Theorem A nonmeager B 2 ω with the Baire Property isn t G 0 -loose. In particular, G 0 isn t Borel-loose. So if an analytic graph G is Borel-loose, it must have countable Borel chromatic number.

31 Borel-looseness The F σ graph D isn t loose, but has Borel chromatic number ℵ 0.

32 Borel-looseness The F σ graph D isn t loose, but has Borel chromatic number ℵ 0. Even worse, the complete bipartite graph with partite sets Q, R \ Q is 0 3 isn t loose and has Borel chromatic number 2.

33 Open Questions

34 Open Questions To what extent can we extend the forcing result more complex graphs?

35 Open Questions To what extent can we extend the forcing result more complex graphs? Is there a closed, non-loose graph that has no perfect cliques?

36 Open Questions To what extent can we extend the forcing result more complex graphs? Is there a closed, non-loose graph that has no perfect cliques? Is there a loose graph with uncountable chromatic number?

37 Thank you!