A Ramsey theorem for metric spaces

Slides from the talk I gave at SetTop2016.

Transcript

1. A Ramsey Theorem for Metric Spaces (”joint work” with Saharon Shelah) Department of Logic FF UK Jonathan Verner
2. Classical Ramsey Theory Department of Logic FF UK Classical Ramsey Theory κ → (λ)ν µ ▶ For any coloring of ν-sized subsets of κ with µ-many colors there is a λ-sized monochromatic subset of κ. ▶ Case ν = 1 is trivial. ▶ What if we add structure?
3. Adding Structure Department of Logic FF UK Ramsey arrow with structure κ →K (λ)1 µ
4. Adding Structure Department of Logic FF UK Ramsey arrow with structure κ →K (λ)1 µ ▶ A class of structures K.
5. Adding Structure Department of Logic FF UK Ramsey arrow with structure κ →K (λ)1 µ ▶ A class of structures K. ▶ For A, B ∈ K a set of embeddings Emb(A, B) of A into B.
6. Adding Structure Department of Logic FF UK Ramsey arrow with structure κ →K (λ)1 µ ▶ A class of structures K. ▶ For A, B ∈ K a set of embeddings Emb(A, B) of A into B. ▶ Arrow holds if for every structure B ∈ K of size κ,
7. Adding Structure Department of Logic FF UK Ramsey arrow with structure κ →K (λ)1 µ ▶ A class of structures K. ▶ For A, B ∈ K a set of embeddings Emb(A, B) of A into B. ▶ Arrow holds if for every structure B ∈ K of size κ,and every partition of B = ∪ α<µ Bα into µ many pieces,
8. Adding Structure Department of Logic FF UK Ramsey arrow with structure κ →K (λ)1 µ ▶ A class of structures K. ▶ For A, B ∈ K a set of embeddings Emb(A, B) of A into B. ▶ Arrow holds if for every structure B ∈ K of size κ,and every partition of B = ∪ α<µ Bα into µ many pieces, there is a structure A ∈ K of size λ and an embedding e ∈ Emb(A, B) of A into B
9. Adding Structure Department of Logic FF UK Ramsey arrow with structure κ →K (λ)1 µ ▶ A class of structures K. ▶ For A, B ∈ K a set of embeddings Emb(A, B) of A into B. ▶ Arrow holds if for every structure B ∈ K of size κ,and every partition of B = ∪ α<µ Bα into µ many pieces, there is a structure A ∈ K of size λ and an embedding e ∈ Emb(A, B) of A into Bsuch that the image e[A] of A is contained in some Bα .
10. Previous Results Department of Logic FF UK Graphs Ordered Graphs Let G be the class of well-ordered undirected graphs and i ∈ Emb(G, H) if i is an order preserving injective mapping and the image of G is an induced subgraph of H graph-isomorphic to G.
11. Previous Results Department of Logic FF UK Graphs Ordered Graphs Let G be the class of well-ordered undirected graphs and i ∈ Emb(G, H) if i is an order preserving injective mapping and the image of G is an induced subgraph of H graph-isomorphic to G. Theorem (Hajnal,Komjáth) 2κ →G (κ)1 κ .
12. Previous Results Department of Logic FF UK Topological Spaces Topological Spaces Let T0 and T1 be the class all T0 and T1 topological spaces, respectively. The set Emb(X, Y) consists of homeomorphic embeddings of a topological space X into Y.
13. Previous Results Department of Logic FF UK Topological Spaces Topological Spaces Let T0 and T1 be the class all T0 and T1 topological spaces, respectively. The set Emb(X, Y) consists of homeomorphic embeddings of a topological space X into Y. Theorem (Nešetřil, Rödl) Let i < 2. Then κµ →Ti (κ)1 µ .
14. Previous Results Department of Logic FF UK Topological Spaces Topological Spaces Let T0 and T1 be the class all T0 and T1 topological spaces, respectively. The set Emb(X, Y) consists of homeomorphic embeddings of a topological space X into Y. Theorem (Nešetřil, Rödl) Let i < 2. Then κµ →Ti (κ)1 µ . Theorem (Weiss) Every T2 topological space can be partitioned into two pieces such that no piece contains a homeomorphic copy of the Cantor set
15. Previous Results Department of Logic FF UK Topological Spaces Topological Spaces Let T0 and T1 be the class all T0 and T1 topological spaces, respectively. The set Emb(X, Y) consists of homeomorphic embeddings of a topological space X into Y. Theorem (Nešetřil, Rödl) Let i < 2. Then κµ →Ti (κ)1 µ . Theorem (Weiss) Every T2 topological space can be partitioned into two pieces such that no piece contains a homeomorphic copy of the Cantor set (under suitable cardinal arithmetic assumptions; these hold e.g. if no measurable cardinals exist).
16. Metric spaces Department of Logic FF UK Metric spaces Let M be the class of metric spaces. For what κ, λ, µ can we have κ →M (κ)1 µ . ▶ By Weiss’ result we need λ < 2ω ▶ We restrict ourselves to bounded metric spaces. ▶ The embeddings will be scaled isometries
17. Metric spaces Department of Logic FF UK Main Theorem Definition A metric space (X, ρ) is bounded if there is d such that ρ(x, y) < d for each (X, ρ). Definition An injective map i : (X, ρ) → (Y, σ) is a scaled isometry if there is ε > 0 such that for each x, y ∈ X we have ρ(x, y) = ε · σ(i(x), i(y))). Let M consist of all bounded metric spaces and Emb(X, Y) be the scaled isometries of X into Y. Theorem (Shelah, V.) 2ω →M (ω)1 ω .
18. Preliminaries Department of Logic FF UK Proof of the main theorem Given a countable bounded metric space (X, ρ) we will find a metric space (Y, d) such that for any partition of Y into countably many pieces one piece contains a scaled isometric copy of X. ▶ By universality of Q ∩ [0, 1] we could only consider X = Q ∩ [0, 1]; this will not be needed.
19. Preliminaries Department of Logic FF UK Proof of the main theorem Given a countable bounded metric space (X, ρ) we will find a metric space (Y, d) such that for any partition of Y into countably many pieces one piece contains a scaled isometric copy of X. ▶ By universality of Q ∩ [0, 1] we could only consider X = Q ∩ [0, 1]; this will not be needed. ▶ It is in fact sufficient to find a pseudometric (Y, ρ) space.
20. Preliminaries Department of Logic FF UK Proof of the main theorem Given a countable bounded metric space (X, ρ) we will find a metric space (Y, d) such that for any partition of Y into countably many pieces one piece contains a scaled isometric copy of X. ▶ By universality of Q ∩ [0, 1] we could only consider X = Q ∩ [0, 1]; this will not be needed. ▶ It is in fact sufficient to find a pseudometric (Y, ρ) space. ▶ Without loss of generality assume that the diameter of X is 1.
21. Preliminaries Department of Logic FF UK Proof of the main theorem Given a countable bounded metric space (X, ρ) we will find a metric space (Y, d) such that for any partition of Y into countably many pieces one piece contains a scaled isometric copy of X. ▶ By universality of Q ∩ [0, 1] we could only consider X = Q ∩ [0, 1]; this will not be needed. ▶ It is in fact sufficient to find a pseudometric (Y, ρ) space. ▶ Without loss of generality assume that the diameter of X is 1.
22. The underlying set Department of Logic FF UK Proof of the main theorem The space Y will consist of some sequences s ∈ <ω1 ω. For a sequence s let π(s) be the largest non-decreasing function bounded by s.
23. The underlying set Department of Logic FF UK Proof of the main theorem The space Y will consist of some sequences s ∈ <ω1 ω. For a sequence s let π(s) be the largest non-decreasing function bounded by s. For r in the range of π(s) let Pr(r, s) = {α < |s| : s(α) = π(s)(α) = r}.
24. The underlying set Department of Logic FF UK Proof of the main theorem The space Y will consist of some sequences s ∈ <ω1 ω. For a sequence s let π(s) be the largest non-decreasing function bounded by s. For r in the range of π(s) let Pr(r, s) = {α < |s| : s(α) = π(s)(α) = r}. Let Y = {s ∈ <ω1 ω : (∀r)(|Pr(s, r)| < ω)}.
25. The (pseudo)metric Department of Logic FF UK Proof of the main theorem Given s ∈ Y and r in the range of π(s) we let X(r, s) = {s ↾ α : s(α) = r}.
26. The (pseudo)metric Department of Logic FF UK Proof of the main theorem Given s ∈ Y and r in the range of π(s) we let X(r, s) = {s ↾ α : s(α) = r}. ▶ define a function f on X(r, s) × X(r, s) ⊆ Y × Y so that it is a suitably scaled isometric copy of an initial segment of X.
27. The (pseudo)metric Department of Logic FF UK Proof of the main theorem Given s ∈ Y and r in the range of π(s) we let X(r, s) = {s ↾ α : s(α) = r}. ▶ define a function f on X(r, s) × X(r, s) ⊆ Y × Y so that it is a suitably scaled isometric copy of an initial segment of X. ▶ extend f to a metric d on Y via the standard ”trick”
28. The (pseudo)metric Department of Logic FF UK Proof of the main theorem Given s ∈ Y and r in the range of π(s) we let X(r, s) = {s ↾ α : s(α) = r}. ▶ define a function f on X(r, s) × X(r, s) ⊆ Y × Y so that it is a suitably scaled isometric copy of an initial segment of X. ▶ extend f to a metric d on Y via the standard ”trick” ▶ prove that d = f where defined
29. The (pseudo)metric Department of Logic FF UK Proof of the main theorem Given s ∈ Y and r in the range of π(s) we let X(r, s) = {s ↾ α : s(α) = r}. ▶ define a function f on X(r, s) × X(r, s) ⊆ Y × Y so that it is a suitably scaled isometric copy of an initial segment of X. ▶ extend f to a metric d on Y via the standard ”trick” ▶ prove that d = f where defined
30. Finding monochromatic copies Department of Logic FF UK Proof of the main theorem Let χ : Y → ω be a partition of Y. Recursively define a sequence ⟨sα : α ≤ γ⟩ such that: 1. s0 = ∅;
31. Finding monochromatic copies Department of Logic FF UK Proof of the main theorem Let χ : Y → ω be a partition of Y. Recursively define a sequence ⟨sα : α ≤ γ⟩ such that: 1. s0 = ∅; and 2. sα+1 = s⌢ α χ(sα);
32. Finding monochromatic copies Department of Logic FF UK Proof of the main theorem Let χ : Y → ω be a partition of Y. Recursively define a sequence ⟨sα : α ≤ γ⟩ such that: 1. s0 = ∅; and 2. sα+1 = s⌢ α χ(sα); and 3. sβ = ∪ α<β sα if β is limit.
33. Finding monochromatic copies Department of Logic FF UK Proof of the main theorem Let χ : Y → ω be a partition of Y. Recursively define a sequence ⟨sα : α ≤ γ⟩ such that: 1. s0 = ∅; and 2. sα+1 = s⌢ α χ(sα); and 3. sβ = ∪ α<β sα if β is limit. Then ▶ γ is a limit ordinal and sγ ̸∈ Y
34. Finding monochromatic copies Department of Logic FF UK Proof of the main theorem Let χ : Y → ω be a partition of Y. Recursively define a sequence ⟨sα : α ≤ γ⟩ such that: 1. s0 = ∅; and 2. sα+1 = s⌢ α χ(sα); and 3. sβ = ∪ α<β sα if β is limit. Then ▶ γ is a limit ordinal and sγ ̸∈ Y ▶ X(r, s) has order typy ω for some r.
35. Finding monochromatic copies Department of Logic FF UK Proof of the main theorem Let χ : Y → ω be a partition of Y. Recursively define a sequence ⟨sα : α ≤ γ⟩ such that: 1. s0 = ∅; and 2. sα+1 = s⌢ α χ(sα); and 3. sβ = ∪ α<β sα if β is limit. Then ▶ γ is a limit ordinal and sγ ̸∈ Y ▶ X(r, s) has order typy ω for some r. ▶ X(r, s) ⊆ Y is a scaled r-colored copy of X.